ACTUARY Actuarial Models 1 — Questions and Answers
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?
- 0.10
- 0.15 (Correct answer)
- 0.20
- 0.25
Correct answer: 0.15
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?
- $9,000
- $9,500 (Correct answer)
- $10,000
- $10,500
Correct answer: $9,500
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?
- $1,040
- $1,080
- $1,120 (Correct answer)
- $1,160
Correct answer: $1,120
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?
- 0.90
- 0.95 (Correct answer)
- 0.99
- 1.00
Correct answer: 0.95
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?
- $5,000 (Correct answer)
- $6,000
- $7,000
- $8,000
Correct answer: $5,000
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 Actuarial Models 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?