Question

Adik and Abang each purchase both a life annuity and a whole life insurance policy on January 1,1980. Each annuity pays $100 per year commencing on January 1,1981. The actuarial present value for the annuity is $1,300 for Abang and $1,370 for Adik. Each insurance policy provides $10,000 payable at the end of the year of death. The actuarial present value for the insurance policy is $3,000 for Adik. Abang is one year older than Adik. Within which of the following ranges is the actuarial present value for Abang?

Answer 1

The actuarially fair premium is calculated for two different risk groups of 50-year-old men, one with a family history of cancer and one without. The premiums differ due to the varying risk of death. Charging a single premium to the entire group could lead to adverse selection.

Step-by-step explanation:

Actuarially Fair Premium Calculation

The question involves calculating the actuarially fair premium for life insurance policies. In the example given, there are two distinct risk groups for 50-year-old men, which significantly affects their insurance premiums due to varying chances of dying within the next year. The concept of adverse selection is critical to understanding insurance markets, as it illustrates the consequences when insurance companies are unable to differentiate between groups based on risk levels.

a. Actuarially Fair Premium for Each Group

To calculate the actuarially fair premium for each group, we consider the expected payout and the probability of death for those with and without a family history of cancer. For the group with a family history of cancer (20% of 1,000 men), the probability of dying in the next year is 1 in 50. The expected payout would be $2,000 (20 men × $100,000 ÷ 50). For the group without a family history (80% of 1,000 men), the probability is 1 in 200, yielding an expected payout of $400 (80 men × $100,000 ÷ 200).

b. Actuarially Fair Premium for the Entire Group

To find the fair premium for the entire group without considering family history, we determine the overall expected payout and divide by the total number of men. The expected payout for the entire group would be $2,400 ($2,000 from those with a family history of cancer and $400 from those without), with a corresponding fair premium of $2.40 per person ($2,400 ÷ 1,000).

c. Consequences of Charging a Single Premium

If an insurer charges a single fair premium to the entire group, there is a risk of adverse selection, where healthier individuals, recognizing that they are subsidizing the higher-risk group, may choose not to purchase insurance. This situation can lead to financial losses for the insurer if only high-risk individuals buy insurance, as they would likely claim much more than what they pay in premiums.