Chapter 5: Problem 29

Expected Value for Life Insurance There is a \(0.9986\) probability that a randomly selected 30 -year-old male lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges \(\$ 161\) for insuring that the male will live through the year. If the male does not survive the year, the policy pays out \(\$ 100,000\) as a death benefit a. From the perspective of the 30 -year-old male, what are the monetary values corresponding to the two events of surviving the year and not surviving? b. If a 30 -year-old male purchases the policy, what is his expected value? c. Can the insurance company expect to make a profit from many such policies? Why?

### Short Answer

## Step by step solution

## - Identify monetary values for survival and death

## - Calculate expected value for the male

## - Determine if the insurance company expects a profit

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### probability

1. **Survival Probability**: The probability that a 30-year-old male will survive the year, given as 0.9986.

2. **Non-Survival Probability**: The probability that the same male will not survive the year, which we calculate as 1 - 0.9986 = 0.0014. Understanding these probabilities helps us later in calculating the expected value and profit for the insurance company.

###### expected value

\[E(X) = p_1 \times x_1 + p_2 \times x_2\]

Where:

**\(p_1\)**: Probability of the first event (surviving the year)**\(x_1\)**: Monetary value if the first event occurs (loss of 161 dollars)**\(p_2\)**: Probability of the second event (not surviving the year)**\(x_2\)**: Monetary value if the second event occurs (gain of 99,839 dollars)

\(p_1 = 0.9986\), \(x_1 = -161\)

\(p_2 = 0.0014\), \(x_2 = 99,839\)

Substitute these values into the formula to calculate:

\[E(X) = 0.9986 \times (-161) + 0.0014 \times 99,839\]

The expected value is \(-20.989\) dollars, indicating an average loss for the 30-year-old male.

###### insurance mathematics

1. **Premium Paid by Male**: When the male survives the year, he loses the insurance premium of 161 dollars. If he does not survive, the insurance pays out 100,000 dollars, less the premium paid, resulting in a 99,839 dollars payout.

2. **Company's Perspective**: For the insurance company, the expected cost per policy is calculated as:

\[0.9986 \times (-161) + 0.0014 \times 99,839 = -20.989\]

The negative expected cost indicates a profit of approximately 21 dollars per policy. This calculation is crucial for the company to set premiums that ensure profitability while offering fair payouts.

###### profit calculation

1. **Premium Collection**: The company collects 161 dollars for each policy sold.

2. **Expected Payout**: The calculated expected cost for the company is \(-20.989\) dollars, which implies that each policy sold results in an average profit of approximately 21 dollars.

Therefore, the mathematical profit for the company per policy is:

\(161 - 20.989 = 140.011\) dollars.

This profit margin ensures the sustainability of the insurance company, allowing it to cover the risks and still remain financially healthy.