MathProblems.info

Problem 153 Solution

E(a) = 200*.01=2
E(a²) = 200²*.01=400
Var(a)=E(a²) - (E(a))² = 400 - 4 = 396

E(b) = 100*.05=5
E(b²) = 100²*.05=500
Var(b)=E(b²) - (E(b))² = 500 - 25 = 475

The variance of total claims is 500*396+300*475 = 340,500
The standard deviation of total claims is 340500^1/2 =~ 583.52

Expected total claims is 500*2 + 300*5 = 2500

Pr(total claims <= total revenue) = .95
Pr(C<=R) = .95 (where C=total claims, R=total revenue)
Pr(C<=R) = .95
Pr(C-m<=R-m) = .95 (where m is the expected total claims)
Pr(C-2500<=R-2500) = .95
Pr((C-2500)/s<=(R-2500)/s) = .95 (where s is the standard deviation)
Pr((C-2500)/583.52<=(R-2500)/583.25) = .95

We can now use the central limit theorem because (C-2500)/583.52 has a normal distribution with mean of 0 and stanrdard deviation of 1.

(R-2500)/583.25 = 1.645
R = 3459.89

So the insurance company needs 3459.89 in revenue. The expected costs of claims is 2500. So k = 3459.89/2500 = 1.38

This is a Society of Actuaries sample problem for course 100.

MathProblems.info home