1. Review of Risk Management Concepts
Section 10
Review of Risk Management Concepts

2. 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

3. 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

4. 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 𝑉𝑎𝑟(𝑋) 𝐸[𝑋] = 𝜎 𝜇

5. 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 𝑌= if 𝑋≤𝑑 𝑋−𝑑 if 𝑋>𝑑
What would the expected payment by the insurer E[Y] be?

6. 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
𝑌= 0 if 𝑋≤𝑑 𝑋 if 𝑋>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’
𝑌= 𝑋≤𝑑 𝑑 ′ ∗ 𝑋−𝑑 𝑑 ′ −𝑑 𝑑<𝑋<𝑑′ 𝑋 𝑋>𝑑′
These are less likely to appear on exam but relatively simple to remember, so it doesn’t hurt to know them
draw Y for disappearing on board

7. Partial insurance – Policy Limit
For a policy limit u, the insurer will only pay an amount up to u when a loss occurs
𝑌= 𝑋 if 𝑋≤𝑢 𝑢 if 𝑋>𝑢
What would E[Y] be in this case?

8. Deductible + Policy Limit
What if you have an insurance policy with both a deductible AND a policy limit?
Policy limit is applied first
𝑌= if 𝑋≤𝑑 𝑋−𝑑 if 𝑑<𝑋≤𝑢 𝑢−𝑑 if 𝑋>𝑢

9. Partial insurance – Proportional Insurance
Specifies a fraction α between 0 and 1, and when a loss occurs, insurer pays αX
𝑌=𝛼∗𝑋
Proportional insurance is not quite as common, but again very easy to remember

10. 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 𝑆= 𝑖=1 𝑛 𝑋 𝑖
Therefore, we can find E[S] and Var[S] by adding up the means and variances of each individual policy (assume independence)

11. 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
𝑃 𝑆≤𝑄 =𝑃 𝑆−𝐸 𝑆 𝑉𝑎𝑟 𝑆 ≤ 𝑄−𝐸 𝑆 𝑉𝑎𝑟 𝑆 = ?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

12. 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?

13. 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: 𝑓 𝑥 = 2 𝑥 3 for 𝑥>1 0 otherwise What is the expected value of the benefit paid under the insurance policy?

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. 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 Calculate the 95th percentile of the insurance company payout on this policy.

20. 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 where 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.

21. 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.

22. Sample Exam #291
A government employee’s yearly dental expense follows a uniform distribution on the interval from 200 to 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).