Suppose a life insurance company sells a ​$270 comma 000 ​1-year term life insurance policy to a 20​-year-old female for ​$290. According to the National Vital Statistics​ Report, 58(21), the probability that the female survives the year is 0.999544. Compute and interpret the expected value of this policy to the insurance company.
Question content area bottom
Part 1
The expected value is ​$
  
enter your response here.
​(Round to the nearest cent as​ needed.)

To calculate the expected value (EV) of the insurance policy to the insurance company, we need to consider the following components:

1. **Premium paid by the policyholder**: This is the amount the company receives, which is $290.
2. **Payout amount**: If the policyholder dies, the insurance company will pay $270,000.
3. **Probability of surviving**: $ P(\text{survive}) = 0.999544 $
4. **Probability of death**: $ P(\text{death}) = 1 - P(\text{survive}) = 1 - 0.999544 = 0.000456 $

### Step-by-step solution:

1. **Expected value if the policyholder survives**:
   - If the policyholder survives, the company keeps the $290 and makes no payout.
   - The expected value contribution from this scenario is $ 290 \times 0.999544 $.

2. **Expected value if the policyholder dies**:
   - If the policyholder dies, the company has to pay out $270,000.
   - The expected value contribution from this scenario is $ (-270,000 + 290) \times 0.000456 $, since they lose $270,000 but keep the $290 premium.

3. **Total expected value**:
   \[
   \text{EV} = (290 \times 0.999544) + ((-270,000 + 290) \times 0.000456)
   \]
   \[
   \text{EV} = (290 \times 0.999544) + (-269,710 \times 0.000456)
   \]

Now let's calculate each part:

- $ 290 \times 0.999544 = 289.868 $
- $ -269,710 \times 0.000456 = -122.897 $

So the total expected value (EV) is:

\[
\text{EV} = 289.868 - 122.897 = 166.97
\]

### Final answer:
The expected value of the policy to the insurance company is **$166.97**.

### Interpretation:
This means that, on average, the insurance company expects to make a profit of $166.97 per policy sold to a 20-year-old female under these conditions.

Would you like further details or have any other questions?

---

**Related questions:**

1. How would the expected value change if the premium were increased to $500?
2. What is the expected value if the probability of survival changes to 0.9997?
3. How does the probability of death affect the overall expected value of the policy?
4. What would the insurance company's expected value be if the payout was reduced to $200,000?
5. How does the company mitigate risk when selling life insurance policies like this?

**Tip**: Expected value helps insurance companies assess profitability by accounting for both potential payouts and premiums collected.

Suppose a life insurance company sells a $270,000 1-year term life insurance policy to a 20-year-old female for $290. According to the National Vital Statistics Report, 58(21), the probability that the female survives the year is 0.999544. Compute and interpret the expected value of this policy to the insurance company.

Learn how to compute the expected value of a $270,000 life insurance policy sold to a 20-year-old female for $290. The problem involves using probability and expected value to determine the insurance company's expected profit. Ideal for high school students in Grades 11-12 studying probability and insurance mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation