An insurance company estimates that the probability that an...

An insurance company estimates that the probability that an individual in a particular risk group will survive one year is $0.99 .$ Such a person wishes to buy a $\$ 75,000$ one-year term life insurance policy. Let $C$ denote how much the insurance company charges such a person for such a policy. a. Construct the probability distribution of $X$. (Two entries in the table will containC.) b. Compute the expected value $E(\boldsymbol{X})$ of $X$. c. Determine the value $C$ must have in order for the company to break even on all such policies (that is, to average a net gain of zero per policy on such policies). d. Determine the value $C$ must have in order for the company to average a net gain of $\$ 150$ per policy on all such policies.

An insurance company estimates that the probability that an individual in a particular risk group will survive
one year is $0.99 .$ Such a person wishes to buy a $\$ 75,000$ one-year term life insurance policy. Let $C$ denote how
much the insurance company charges such a person for such a policy.
a. Construct the probability distribution of $X$. (Two entries in the table will containC.)
b. Compute the expected value $E(\boldsymbol{X})$ of $X$.
c. Determine the value $C$ must have in order for the company to break even on all such policies
(that is, to average a net gain of zero per policy on such policies).
d. Determine the value $C$ must have in order for the company to average a net gain of $\$ 150$ per
policy on all such policies.

Key Concepts

Premium Setting for Desired Profit

Beyond breaking even, insurers often aim for a desired profit margin per policy. Setting premiums requires adjusting the charged amount so that, after accounting for the probabilities and costs of claims, the expected net gain meets a specified profit target. This process is fundamental for maintaining a profitable portfolio while managing risk.

Random Variable

A random variable in this context is a numerical representation of the net gain or loss associated with an insurance policy. It encapsulates the possible outcomes—such as receiving a premium with no payout or paying out a benefit after a claim—and assigns a monetary value to each, which is essential to quantify and manage risk.

Discrete Probability Distribution

A discrete probability distribution provides a description of the probability associated with each possible value of the random variable. In insurance, this is used to map out the likelihood of different events, such as the insured surviving or dying, and helps in evaluating the overall risk profile of a policy by assigning probabilities to each outcome.

Expected Value

The expected value is the weighted average of all possible outcomes of a random variable, calculated by multiplying each outcome by its probability of occurrence and summing the results. In insurance, this concept is crucial for determining whether the premiums charged will suffice to cover potential claims, thus guiding decisions on setting rates to ensure financial viability.

Break-Even Analysis

Break-even analysis in an insurance context involves setting parameters such that the expected net gain from premiums and payouts equals zero. This principle is used to calculate the minimal premium required to cover expected losses, ensuring that the insurer does not incur a loss on average.

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