A life insurance company sells a term insurance policy to a...

A life insurance company sells a term insurance policy to a 21-year-old male that pays $\$ 100,000$ if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $\$ 250$ each year as payment for the insurance. The amount Y that the company earns on this policy is $\$ 250$ per year, less the $\$ 100,000$ that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$ profit earned by the company: Age at death: 21 22 23 24 25 26 or more Profit: ?$99,750 ?$99,500 Probability: 0.00183 0.00186 0.00189 0.00191 0.00193 (a) Copy the table onto your paper. Fill in the missing values of $Y .$ (b) Find the missing probability. Show your work. (c) Calculate the mean $\mu_{Y}$ . Interpret this value in context.

A life insurance company sells a term insurance policy to a 21-year-old male that pays $\$ 100,000$ if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $\$ 250$ each year as payment for the insurance. The amount Y that the company earns on this policy is $\$ 250$ per year, less the $\$ 100,000$ that it must pay if the insured dies. Here is a partially completed table that shows information about risk of mortality and the values of $Y=$ profit earned by the company:

Age at death: 21 22 23 24 25 26 or more
Profit: ?$99,750 ?$99,500
Probability: 0.00183 0.00186 0.00189 0.00191 0.00193

(a) Copy the table onto your paper. Fill in the missing values of $Y .$
(b) Find the missing probability. Show your work.
(c) Calculate the mean $\mu_{Y}$ . Interpret this value in context.

Key Concepts

Term Insurance Policy

A term insurance policy is a type of life insurance that provides coverage for a specific period. It pays a death benefit if the insured passes away during the term of the policy. This concept is crucial when evaluating the risk and return for both the insurer and the policyholder, as it defines the period during which the coverage is active and indicates that no benefits are paid if the insured survives the term.

Annual Premium Payments

Premiums are the regular payments made by the policyholder to maintain the insurance coverage. In evaluating insurance products, understanding premium structure is key because it directly affects the insurer’s income stream. These payments are typically collected periodically, and in the context of the problem, they contribute to the company’s profit until a potential payout is required upon the policyholder's death.

Mortality Tables

Mortality tables, or life tables, are statistical tools that provide the probabilities of death for individuals at various ages. They are essential in insurance as they help insurers assess risk and determine appropriate premium levels. Mortality tables allow for the estimation of the likelihood that an insured event, such as death within a specific period, will occur, which in turn influences pricing and reserving decisions.

Profit or Loss Random Variable

In the insurance context, the profit (or loss) random variable represents the financial outcome from a policy after accounting for both premiums received and benefits paid out. This variable is defined for each possible event (such as death occurring at different ages or not at all) and serves as the basis for risk assessment and pricing. Calculating the values of this random variable under different scenarios provides insight into the insurer's financial exposure.

Expected Value Calculation

The expected value of a random variable is a fundamental concept in probability and statistics that represents the weighted average of all possible outcomes, with the weights being the probabilities of those outcomes. In insurance, computing the expected value of a profit or loss variable helps a company understand its average earnings over many policies. It is crucial for pricing strategies and ensuring that premiums are set at a level that is likely to cover expected payouts plus expenses.

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