Review of Risk Management Concepts - PowerPoint Presentation

Slide1

Review of Risk Management Concepts

Section 10Slide2

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 contractSlide3

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 premiumSlide4

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

 Slide5

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?

 Slide6

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 themSlide7

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?

 Slide8

Deductible + Policy Limit

What if you have an insurance policy with both a deductible AND a policy limit?

Policy limit is applied first

 Slide9

Partial insurance – Proportional Insurance

Specifies a fraction

α

between 0 and 1, and when a loss occurs, insurer pays

α

X

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

 Slide10

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

. X

iThe 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)

 Slide11

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

 Slide12

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?Slide13

Sample Exam #53

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

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

 Slide14

Sample Exam #85

The total claim amount for a health insurance policy follows a distribution with density function f(x) = 1/1000 *

exp

(-x/1000), x>0.

The premium for the policy is set at the expected total claim amount plus 100.

If 100 policies are sold, calculate the approximate probability that the insurance company will have claims exceeding the premiums collected.Slide15

Sample Exam #127

The amounts of automobile losses reported to an insurance company are mutually independent, and each loss is uniformly distributed between 0 and 20,000. The company covers each such loss subject to a deductible of 5,000.

Calculate the probability that the total payout on 200 reported losses is between 1,000,000 and 1,200,000.Slide16

Sample Exam #161

An auto insurance policy has a deductible of 1 and a maximum claim payment of 5. Auto loss amounts follow an exponential distribution with mean 2.

Calculate the expected claim payment made for an auto loss.Slide17

Sample Exam #324

The independent random variables X and Y have the same mean. The coefficients of variation of X and Y are 3 and 4 respectively.

Calculate the coefficient of variation of (X+Y)/2.Slide18

Sample Exam #147

The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%.

Calculate the percentage reduction on the variance of the claim payment.Slide19

Sample Exam #150

An automobile insurance company issues a one-year policy with a deductible of 500. The probability is 0.8 that the insured automobile has no accident and 0.0 that the automobile has more than one accident. If there is an accident, the loss before application of the deductible is exponentially distributed with mean 3000.

Calculate the 95

th

percentile of the insurance company payout on this policy.Slide20

Sample Exam #167

Damages to a car in a crash are modeled by a random variable with density function

f(x) = c(x^2 – 60x + 800), 0<x<20; 0, otherwise

w

here c is a constant

A particular car is insured with a deductible of 2. This car was involved in a crash with resulting damages in excess of the deductible.

Calculate the probability that the damages exceeded 10.Slide21

Sample Exam #287

The loss L due to a boat accident is exponentially distributed.

Boat insurance policy A covers up to 1 unit for each loss. Boat insurance policy B covers up to 2 units for each loss.

The probability that a loss is fully covered under policy B is 1.9 times the probability that it is fully covered under policy A.

Calculate the variance of L.Slide22

Sample Exam #291

A government employee’s yearly dental expense follows a uniform distribution on the interval from 200 to 1200. The government’s primary dental plan reimburses an employee for up to 400 of dental expense incurred in a year, while a supplemental plan pays up to 500 of any remaining dental expense.

Let Y represent the yearly benefit paid by the supplemental plan to a government employee.

Calculate

Var

(Y).