Question

An obstetrics specialist (baby deliverer) faces a small but nonzero chance of being sued for negligence. You estimate that the chance of being sued in any year is 5%. [If the client is sued] you estimate that there is a 30% chance the client would win the case with costs awarded, a 40% chance the client would win the case but have to pay legal fees of $100,000 and a 30% chance the client would lose the case and have to pay a total of $1,000,000, including all legal fees.

(a) How much would a reasonable yearly insurance premium be?

(b) Pegasus Insurance adjustors are worried about the increasing trend in litigation. They think that the probability of being sued could be as high as 10%. What would an appropriate premium be?

(c) The adjustors also estimate that payouts when clients lose are more likely to be $5,000,000 than $1,000,000 and want to insure on this basis. How much will the premium be?

(d) Pegasus researchers estimate that orthopedic specialties have the following conditions: (1) the chance of being sued is 7%, (2) the chance of winning but without costs awarded is 35% [legal fees ($100,000) are the insurance companies’ responsibility], (3) the chance of losing is 35%, and (4) the amount awarded when a case is lost averages $2,000,000. What is a fair premium for this specialty?

Answer 1

a) Expected premium = amount to pay in case sued/total number of insurance

--> if 100 users are buying premium, 5 will sue (5% is the chance of being sued)

--> Of 5%, 30% (i.e. 1.5% of overall) are win with cost awarded, hence, no expense for insurance provider. (Case - A)

--> Of 5%, 40% (i.e. 2% of overall) results in paying $100,000. (Case - B)

--> Of 5%, 30% (i.e. 1.5% of overall) results in paying $1,000,000. (Case - C)

Total Expected expense = 2*100,000 + 1.5*1,000,000 = $1,700,000

Premium= 1,700,000/100 = $17,000

b) If change of being sued doubles (5% to 10%) with everything else constant, premium will also increase by same proportion i.e it will also double to $34,000.

c) The C in (a) would result in payments of $5,000,000 and not $1,000,000. This will change the expected expense.

New Expected expense = 2*100,000 + 1.5*5,000,000 = $7,700,000

Premium = 7,700,000/100 = $77,000

d) Below are the values as per the new conditions -

--> if 100 users are buying premium, 7 will sue (7% is the chance of being sued)

--> Of 7%, 30% (i.e. 2.1% of overall) are win with cost awarded, hence, no expense for insurance provider. (Case - A)

--> Of 7%, 35% (i.e. 2.45% of overall) results in paying $100,000. (Case - B)

--> Of 7%, 35% (i.e. 2.45% of overall) results in paying $2,000,000. (Case - C)

New Expected expense = 2.45*100,000 + 2.45*2,000,000 = $5,145,000

Premium = 5,145,000/100 = $51,450