ACTUARY Actuarial Models

Question 1

The probability that a person aged 40 survives to age 50 is 0.85. What is the probability that the same person dies before age 50?

Accepted Answer

0.15

Answer

The probability of death is the complement of the survival probability:
P(death before 50)=1−P(survival to 50)=1−0.85=0.15

Question 2

An insurance company covers losses up to $10,000 per claim. If the probability distribution of losses is:
P(X≤10,000)=0.7, P(X>10,000)=0.3,
and the average claim for losses exceeding $10,000 is $15,000, what is the expected claim amount?

Accepted Answer

$9,500

Answer

The expected claim amount is calculated as:
E(X)=P(X≤10,000)⋅10,000+P(X>10,000)⋅15,000
E(X)=(0.7⋅10,000)+(0.3⋅15,000)=7,000+4,500=9,500

Question 3

An insurance company observes that the average claim amount for a policyholder is $1,200, based on past experience. The population mean is $1,000 with a variance of $400. What is the Bayesian estimate of the claim amount if the weight of credibility (Z) is 0.6?

Accepted Answer

$1,120

Answer

The Bayesian estimate is a weighted average of the policyholder's mean and the population mean:
Bayesian Estimate=Z⋅Policyholder Mean+(1−Z)⋅Population Mean
Bayesian Estimate=(0.6⋅1,200)+(0.4⋅1,000)=720+400=1,120

Question 4

The hazard rate for a survival model is given by ℎ(𝑡)=0.02. What is the probability that an individual survives beyond time t=5?

Accepted Answer

0.95

Answer

The survival probability is related to the hazard rate through the cumulative hazard function:
S(5)=e −0.1 ≈0.95

Question 5

An insurance company models aggregate losses as the sum of 𝑁 independent claims, where 
𝑁 follows a Poisson distribution with mean 𝜆=5, and each claim amount is $1,000. What is the expected aggregate loss?

Accepted Answer

$5,000

Answer

E(Aggregate Loss)=E(N)⋅E(Claim Amount)
Here, 

Here, 
E(N)=λ=5E(Claim Amount)=1,000:
E(Aggregate Loss)=5⋅1,000=5,000

Actuary Certification Practice Test

Actuary Certification Practice Test

ACTUARY Actuarial Models