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Expected Value: Life Insurance Jim is a 60 -year-old Anglo male in reasonably good health. He wants to take out a \(\$ 50,000\) term (that is, straight death benefit) life insurance policy until he is \(65 .\) The policy will expire on his 65 th birthday. The probability of death in a given year is provided by the Vital Statistics Section of the Statistical Abstract of the United States (116th Edition). \begin{tabular}{l|ccccc} \hline\(x=\) age & 60 & 61 & 62 & 63 & 64 \\ \hline\(P\) (death at this age) & \(0.01191\) & \(0.01292\) & \(0.01396\) & \(0.01503\) & \(0.01613\) \\ \hline \end{tabular} Jim is applying to Big Rock Insurance Company for his term insurance policy. (a) What is the probability that Jim will die in his 60 th year? Using this probability and the \(\$ 50,000\) death benefit, what is the expected cost to Big Rock Insurance? (b) Repeat part (a) for years \(61,62,63\), and 64 . What would be the total expected cost to Big Rock Insurance over the years 60 through \(64 ?\) (c) Interpretation If Big Rock Insurance wants to make a profit of \(\$ 700\) above the expected total cost paid out for Jim's death, how much should it charge for the policy? (d) Interpretation If Big Rock Insurance Company charges \(\$ 5000\) for the policy, how much profit does the company expect to make?

Step by step solution

01

Calculating Expected Cost for Age 60

The probability of Jim dying at age 60 is \( P(60) = 0.01191 \). The expected cost to Big Rock Insurance is \( 0.01191 \times 50,000 = \$595.50 \).

02

Calculating Expected Cost for Age 61

The probability of Jim dying at age 61 is \( P(61) = 0.01292 \). The expected cost to Big Rock Insurance is \( 0.01292 \times 50,000 = \$646.00 \).

03

Calculating Expected Cost for Age 62

The probability of Jim dying at age 62 is \( P(62) = 0.01396 \). The expected cost to Big Rock Insurance is \( 0.01396 \times 50,000 = \$698.00 \).

04

Calculating Expected Cost for Age 63

The probability of Jim dying at age 63 is \( P(63) = 0.01503 \). The expected cost to Big Rock Insurance is \( 0.01503 \times 50,000 = \$751.50 \).

05

Calculating Expected Cost for Age 64

The probability of Jim dying at age 64 is \( P(64) = 0.01613 \). The expected cost to Big Rock Insurance is \( 0.01613 \times 50,000 = \$806.50 \).

06

Total Expected Cost Over Ages 60 to 64

Adding the expected costs for ages 60 to 64 gives: \( \\(595.50 + \\)646.00 + \\(698.00 + \\)751.50 + \\(806.50 = \\)3497.50 \).

07

Determining Charge for Profit Margin

To make a profit of \\(700, Big Rock Insurance should charge \\)3497.50 (total expected cost) + \\(700 (desired profit) = \\)4197.50.

08

Calculating Expected Profit at $5000 Charge

If the policy is charged at \\(5000, then the expected profit is \\)5000 - \\(3497.50 = \\)1502.50.

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Key Concepts

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

Understanding Probability in Life Insurance

Probability is a crucial concept in life insurance. It helps insurance companies determine the likelihood of events, like a policyholder's death within a specified timeframe. In Jim's case, the probability of him dying in each year from age 60 to 64 is calculated using statistical data. - **Why Probability Matters:** It is the foundation upon which insurance companies base their premiums and benefits calculations. By estimating how likely Jim is to pass away each year, Big Rock Insurance can predict the expected payout.- **Probability Values:** For instance, the probability of Jim's death at age 60 is 0.01191. If we multiply this by the insurance payout (\(0.01191 \times 50,000 = \) $595.50), we get an idea of the financial obligation the company might face that year.This approach ensures that the company can cover its commitments while maintaining profitability. Over several years, these individual probabilities paint a picture of Jim's expected life events, guiding both insurance pricing and risk management.

The Role of Life Insurance

Life insurance serves as a financial safety net, offering peace of mind to policyholders and their families. For Jim, a term life insurance policy provides coverage that only pays a death benefit if he dies within a specific period. - **Term Insurance:** This type of policy is temporarily active, in Jim's case from age 60 to 64. It is different from permanent life insurance because it is simpler and generally less expensive. - **Death Benefit:** This is a predetermined amount of money paid to beneficiaries after the policyholder’s death. Here, Jim’s policy offers a death benefit of $50,000. Understanding how a policy functions helps policyholders make sound decisions. They must balance the likelihood of needing insurance with its cost. Insurers use life tables and statistical models to establish these provisions to balance affordability and protection effectively.

Statistical Analysis in Insurance

Statistical analysis uses mathematical principles to study and interpret data, providing insights and predictions. Insurance companies rely heavily on these techniques to make data-driven decisions. - **Calculated Expectations:** By integrating probability with financial impacts, insurers calculate expected costs for potential events. This approach clarifies the risk associated with each policy. - **Cumulative Analysis:** For Jim, the cumulative expected cost for his term policy is the sum of expected costs from age 60 to 64. It totals to $3,497.50, reflecting the company's obligation if stats remain consistent. Using statistical analysis, insurers not only assess risks but also formulate pricing strategies. Companies need to earn profits, so they incorporate desired margins into premium prices. For Jim's insurer, calculating a desired profit and charging accordingly is crucial. The application of statistical analysis ultimately shapes the financial health and competitiveness of an insurance firm.