Review of Risk Management Concepts

Section 10

Loss distributions and insurance

An insurance policy is a contract between the party that is at risk (the policyholder) and the insurer

The policyholder pays a premium to the insurer

In return the insurer reimburses certain claims to the policyholder A claim is all or part of the loss,

depending on contract

Modeling a loss random variable Unless indicated otherwise, assume the

amount paid to the policyholder is equal to the amount of the loss (“full insurance”)

The random variable X represents the amount of the loss Don’t forget to include 0 as an outcome for X – if

no loss occurs E[X] is then the expected claim on the

insurer It is also called the pure premium – if no

administrative or other costs are factored in, it would be how much the company asks for as a premium

Modeling a loss random variable

E[X] is the pure premium

Var[X] is another measure of risk

The unitized risk or coefficient of variation is

Partial Insurance - Deductibles For a deductible amount = d, the

policyholder pays for all losses less than d This means the insurer pays nothing when

loss X < d, and pays the difference when X > d

The amount Y paid by the insurer can be described as

What would the expected payment by the insurer E[Y] be?

Variations on deductibles Franchise deductible

Insurer pays 0 if loss is below d but pays full amount of loss X if the loss if above d

Disappearing deductible has lower limit d and upper limit d’ Deductible amount reduces linearly

from d to 0 as loss increases from d to d’

These are less likely to appear on exam but relatively simple to remember, so it doesn’t hurt to know them

Partial insurance – Policy Limit

For a policy limit u, the insurer will only pay an amount up to u when a loss occurs

What would E[Y] be in this case?

Deductible + Policy Limit

What if you have an insurance policy with both a deductible AND a policy limit? Policy limit is applied first

Partial insurance – Proportional Insurance

Specifies a fraction α between 0 and 1, and when a loss occurs, insurer pays αX

Proportional insurance is not quite as common, but again very easy to remember

The Individual Risk Model This models the aggregate claims in a

portfolio of insurance policies Assume the portfolio consists of n policies

with the claim for policy i being the r.v. Xi

The aggregate claim is the random variable S

Therefore, we can find E[S] and Var[S] by adding up the means and variances of each individual policy (assume independence)

Normal Approximation to Aggregate Claims

For the aggregate distribution S, if we know E(S) and Var(S), we can approximate probabilities for S with the normal distribution ?th percentile

For example, if insurer collects premium Q, there is a ?% chance that aggregate claims will be less than the premium collected Questions like this are frequent

Mixture of Loss Distributions

This is just the application of what we learned in section 9

For example, if there are high, medium, and low classes of policyholders with different distributions, you may be asked to describe the distribution of a random loss from the portfolio

Sample Exam #48

An insurance policy on an electrical device pays a benefit of 4000 if the device fails during the first year. The amount of the benefit decreases by 1000 each successive year until it reaches 0. If the device has not failed by the beginning of any given year, the probability of failure during that year is .4.

What is the expected benefit under this policy?

Sample Exam #53

An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder’s loss, Y, follows a distribution with density function:

What is the expected value of the benefit paid under the insurance policy?

Sample Exam #56

An insurance policy is written to cover a loss, X, where X has a uniform distribution on [0,1000].

At what level must a deductible be set in order for the expected payment to be 25% of what it would be with no deductible?