A model for Insurance Losses

Our first insurance model: a mathematical model for insurance losses.

Firstly, some notation:
E(X) = The expected value of X
Var(X) = The variance of X
StDev(X) = The standard deviation of X
SQRT(X) = The square root of X

Individual Insurance Losses

Let's use car insurance as an example because a lot of people have had first hand experience with it. Most Australians are forced to drive and hence reluctantly, experience auto insurance.

Suppose that there is a 70% percent chance that I will not make an insurance claim; presumably because I will not have a car accident. Then say there is a 20% chance that I will make an insurance claim for something small: $700, something mostly cosmetic and perhaps only involving my car.

Suppose that there is a 5% chance that I will make a $6000 insurance cliam.
A 3% chance that I will cause some appreciable damage to the tune of $18,000.
A 1.5% chance that I will write-off two moderately priced cars for a total of $60,000.
Finally, a 0.5% chance of causing a catastrophic accident with a $350,000 damage bill.

Suppose that the amount the insurance company must pay (the insurance loss) is a random variable, X. From the above, assuming that the insurance excess is zero, we can deduce a discrete distribution for X, f(x).

The discrete distribution for insurance payout is above. The probability of the payout is on the left and the amount of the payout is on the right.

So E(X) = .7*0+.2*700+.05*6000+.03*18000+.015*60000+.005*350000=$3,630

If there is an insurance excess, you can subtract the excess from the insurance payout.

Homework Exersises:
1. Find Var(X) and the standard deviation of X.
2. Suppose that there is an insurance excess of $500. What would the mean insurance payout and its standard deviation be then?


The distribution of insurance losses (insurance payouts) can also be continuous. However, that will be covered in a later post.

Collective insurance payouts
The random variable X is the insurance payout for one individual. Now the expected insurance payout for all individuals/customers in a given time period is the sum of their indiviual means. In this post, we will assumne that everyone has the same insurance payout distribution. We will denote the total/collective insurance payout as Y.

If we explicitly assume that all X's are independant:
E(Y) = the sum of the insurance payouts for each insurance customer.

Suppose that there are n insurance customers. Var(Y) = n*Var(X). (Remember that we are assuming that every insurance customer has the same distribution of X.)

Then StDev(Y) = StDev(X)*SQRT(n)

The standard deviation is a measure of risk. For a large n, StDev(Y) is much less than sum of StDev(X).

Bibliography:
Course notes for MTH3251/ETC3510/ETC5351 at Monash University by Fima Klebaner Semester 1, 2009.