BILL ADONGO

CENTRAL AGGREGATE LOSS MODEL

Bill Adongo’s Manuscript:

“In Actuarial practices, aggregate loss distribution is used extensively in all practical fields such as ratemaking, premium making and developing of loss distributions and models. Aggregate models include portfolio aggregate loss models, aggregate variance methods, fast fouries transformation, aggregate life model etc. to open the development of the central aggregate loss models I must start with the central variance method first.

Theorem(1):

If the pgf of the secondary distribution of S is var(s_i)=∑µ_iσ_i, the distribution S_i,….,S_n would have independent compound distribution X~S_n[p(t)] with µ_i=[1-p(t)]/p(t) and σ_i=[1-p(t)]/p(t)². Also, the distribution S_i,….., S_n would have independent compound distribution with µ_i²=var(s)/2σ_n and σ_i=Var(s)/2µ_n.

Example(1):

Let var(s)=88, and X~S_n(0.4). Compute µ_x, σ_x² and σ_n, µ_n.

SOLUTION

µ_i=[1-p(t)]/p(t)=0.6/0.4=1.50

σ_i=[1-p(t)]/p(t)²=0.6/0.42=1.43

σ_n=var(s)/2µ_x²=88/2(1.52)=19.56

µ_n=var(s)/2√σ_x=88/2√1.43=36.79

Theorem(2):

If the pgf of the secondary distribution of S is p(t)=∑w_ip(t) where w_i=λ_i/λ, the distribution S_i,…., S_n would have independent compound posson distribution with parameters λ=λ_i and p(t)=p(t)/2w_i.

Theorem(3):

Given the secondary distribution which has expectation of x is E(X)=∑p_iµ_i where p_i=E(X)/2µ_i, the distribution of the variance would expressed us:

Var(x)=∑E(X)/2µ_i[(µi-µ)²+σ²]

Example(2):

The claim frequency of a bad driver is distribution at PN(4), and the claim frequency of a good driver is distributed as PN(1), If the expected value is 1.6;  what is the probabilities and variance of the good and bad driver.

P_good=E(X)/2µ_good=1.6/2(1)=0.8

P_good=E(x)/2µ_bad=1.6/2(4)=0.2

Var(x)=1.6/2(4)[(4-1.6)2+4]+1.6/2(1)[(1-1.6)2+1]

Var(x)=0.2X9.76+0.8X1.36

Var(x)=3.04”