Aggregate Loss Guided Tutorials

Consider the collective risk model, which has the representation;

$$S=X_1+…+X_N$$

Where (S) the aggregate loss of random number (N) of individual claims (X_1,…,X_N). Here, we assume that conditional on (N=n) , (X_1,…,X_n) are (i.i.d) random variables. Also, the frequency (N) and severity (Y) are independent

Using the law of iterated expectations, the mean of (S) is given by;

$$begin{array}{ll}
E(S)
&=E[E(S|N)]=E[NE(X)]=E(N)E(X) \
&=mu E(N)quad quad end{array}$$

Where (mu=E(X_i)).

Using the law of total variation, the variance of (S) is given by;
$$begin{array}{ll}
Var(S)
&=E[Var(S|N)]+Var[E(S|N)] \
&=E[N Var(X)]+Var[N E(X)] \
&=E(N)Var(X)+Var(N)E(X)^2 \
&=sigma^2E(N)+mu^2Var(N) quad quad end{array}$$

Where (Var(X_i)=sigma^2).

Special case: Poisson distributed frequency. (Nsim Poisson(lambda))
$$begin{array}{ll}
E(N)&=Var(N)=lambda \
Var(S)&=lambda(sigma^2+mu^2)=lambda E(X^2)
end{array}$$

The pdf(pmf) of (S) is given by;

$$f_S(s)=sum_{n=0}^{infty}Pr(N=n)cdot f_X^{*n}(s) quad quad (1)$$

This means that to calculate (f_S(s)), we must sum up over all possibilities of (n) claims summing up to (s). The n-fold convolution of (f_X) is (f_X^{*n}) and (f_X^{*0}(0)=1). To obtain the cdf of (S), its given by;

$$F_S(s)=sum_{i=0}^{s}f_S(i)quad quad (2)$$

To apply these rules,

• From the question, identify the distribution of the claim frequency (N) and severity (X).
• If the question requires you to calculate the pdf(pmf) of (S) ,(f_S(s)), its is given by equation (1) above.
• If the question requires you to calculate the cdf, (F_S(s)) it is given by equation (2).

When the number of observed losses (n),is large, the observed aggregate losses are approximated normally distributed. To use normal approximation for aggregate loss distributions, (F_S(s)), we use the formula below:
$$F_S(s)=Pr(S leq s)=Phileft(frac{s-mu_s}{sigma_s}right) quad quad (1)$$

Where;
$$mu_s=E(S) quad text{and} quad sigma_s=sqrt{Var(S)}$$

When (n) is small then the distribution of the aggregate losses is positively skewed and hence its better to approximate aggregate losses with a distribution that is positively skewed (e.g LogNormal distribution ). To use LogNormal approximations:
$$F_S(s)=Pr(S leq s)=Pr(ln S leq ln s)=Phileft(frac{ln s-mu_s}{sigma_s}right) quad quad (2)$$

Where (u_s) and (sigma_s) are found using method of moments :

$$
left{begin{array}{ll}
exp{mu_s+sigma_s^2/2}=E(S)\
exp{2mu_s+2sigma_s^2}=E(S^2)=E(S)^2+Var(S) end{array}right. $$

To apply these rules,

• From the question, identify the distribution of the claim frequency (N) and severity (X).
• Calculate (E(S)) and (Var(S)) using the appropriate formulae
• Using equation (1) compute the aggregate loss distribution function (F_S(s)).
• When (n) is small,use equation (2) for a LogNormal approximation for the aggregate loss distribution function (F_S(s)) .

The stop loss premium is the expected aggregate losses above of aggregate deductible. Here, aggregate loss means the sum of individual claims payments after individual claim modifications such as policy limits and deductibles but before aggregate modifications. An aggregate deductible is a deductible applied to aggregate losses rather than individual losses.

Assuming the frequency (N) and severity (X) are independent, the mean of aggregate losses,(S) , is given by;
$$E(S)=E(N)E(X) quad quad (1)$$

With an aggregate deductible (d), the stop loss premium is the mean aggregate loss (E(S)) minus the expected value of the aggregate loss limited to (d) , (E(Swedge d)). The stop loss premium is given by :
$$E[(S-d)_+]=E(S)-E(Swedge d) quad quad (2)$$

For discrete aggregate loss distributions:
$$E(Swedge d)=sum_{slt d} s f_S(s) + d (1-P(Slt d)) $$

For continuous aggregate loss distributions:
$$E(Swedge d)=int_0^d s f_S(s) ds + int_d^{infty} d f_S(s) ds \
= int^d_0 (1-F(s)) ds . $$

To apply these rules,

• From the question, identify the distribution of the claim frequency (N) and severity (X).
• Calculate the mean aggregate loss (E(S)) and the expected value of the aggregate loss limited to (d), (E(Swedge d)), using the appropriate formulae
• Using equation (2) compute the stop loss premium (E[(S-d)_+]) .

Consider an insurance contract with deductible d. This causes the number of claims for amounts greater than zero to change. Let (N^L) be the number of losses and given (X_i), the ground-up losses for (i=1,…,n^L), denote the number of claims by (N^P) . We assume , given (N^L=n^L) , (X_i) are (i.i.d) with common distribution (X). Then;

$$N^P=I(X_1 gt d)+…+I(X_{N^L}gt d)$$

It can be shown that for distributions of the ((a,b,0)) class (N_L) and (N_P) follow the same distribution but with different parameters. Let (v=Pr(X gt d)), then we have:

$$
{scriptsize
begin{matrix}
begin{array}{lll}
hline
text{Distribution} & N^L & N^P \
hline
text{Poisson} & lambda & vlambda\
text{Binomial} & n, p & n, vp\
text{Negative binomial} & r,beta & r, vbeta \
hline \
end{array}
end{matrix}
}
$$

Also, recall that (X^L) , the random variable for the payment per loss with deductible (d) is given by :
$$X^L=
left{begin{array}{ll}
0 & X_i lt d \
X_i-d & X_i geq d end{array} right. $$

And (X^P),the random variable for the per payment on an insurance with deductible (d) is the payment per loss variable conditioned on (X>d) or

$$X^P=
left{begin{array}{ll}
undefined & X_i lt d \
X_i-d & X_i geq d end{array} right. $$

In modeling aggregate losses with per loss deductibles, one way is to model the number of losses (N^L) in which no modification is needed for the frequncy model and use the payment per loss variable ,(X^L). Another way is to use the modified frequency model (N^P) with the per payment random variable (X^P). Then, we can express the aggregate loss below:

$$begin{array}{ll}
S
&=X_1^L+…+X_{N^L}^L\
&=X_1^P+…+X_{N^P}^P
end{array}$$

If (X) follows uniform, exponential or Pareto then using the per payment approach makes calculation easier.

To apply these rules,

• From the question, identify coverage modification, the distribution of the claim frequency (N) and severity (X).
• Model the aggregate claims using the per-loss apporach or using the per-payment approach