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Expected Value: Life Insurance Sara is a 60 -year-old Anglo female in reasonably good health. She wants to take out a \(\$ 50,000\) term (that is, straight death benefit) life insurance policy until she is \(65 .\) The policy will expire on her 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) & \(0.00756\) & \(0.00825\) & \(0.00896\) & \(0.00965\) & \(0.01035\) \\ \hline \end{tabular} Sara is applying to Big Rock Insurance Company for her term insurance policy. (a) What is the probability that Sara will die in her 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 Sara'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

Calculate Expected Cost for Age 60

To find the expected cost for age 60, multiply the probability of death at age 60 by the death benefit. \( E(60) = P(60) \times \text{Benefit} = 0.00756 \times 50,000 = 378\). This means the expected cost for Sara's death at age 60 is \$378.

02

Calculate Expected Costs for Ages 61, 62, 63, and 64

Repeat the calculation for ages 61 to 64:- Age 61: \( E(61) = 0.00825 \times 50,000 = 412.5\)- Age 62: \( E(62) = 0.00896 \times 50,000 = 448\)- Age 63: \( E(63) = 0.00965 \times 50,000 = 482.5\)- Age 64: \( E(64) = 0.01035 \times 50,000 = 517.5\).

03

Calculate Total Expected Cost

Add the expected costs for ages 60 through 64 to find the total expected cost: \( 378 + 412.5 + 448 + 482.5 + 517.5 = 2238.5\). The total expected cost to Big Rock Insurance over the 5 years is \$2238.50.

04

Determine Policy Charge for Desired Profit

Big Rock Insurance wants to make a profit of \\(700 over the expected cost. Therefore, charge: \( \text{Policy Charge} = \text{Total Expected Cost} + \text{Desired Profit} = 2238.5 + 700 = 2938.5\). The company should charge \\)2938.50 for the policy.

05

Calculate Expected Profit with Different Policy Charge

If the policy is charged at \\(5000, calculate the expected profit:\( \text{Expected Profit} = \text{Policy Charge} - \text{Total Expected Cost} = 5000 - 2238.5 = 2761.5\). The expected profit is \\)2761.50.

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Key Concepts

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

Understanding Probability Theory

Probability theory is the mathematical framework for quantifying uncertainty. In simple terms, it helps us determine how likely an event is to occur. In scenarios like life insurance, probability theory is crucial as it allows us to estimate the likelihood of events such as death, which is inherently uncertain.

To calculate the probability of an event, like the death of Sara at a certain age, probabilities are assigned to possible outcomes based on historical data or statistical analysis. These probabilities help insurers determine risk and, subsequently, the expected cost they might incur because of that risk.

The expected value is a key concept, representing the average outcome of all possible events, weighted by their probabilities. For instance, if the probability of Sara's death at age 60 is 0.00756, the expected cost for the insurer is \( E = P \times ext{Benefit} \). Therefore, insurance companies use probability to make informed decisions about policy costs and coverage.

Life Insurance Basics

Life insurance involves a contract between an individual and an insurance company, where the insurer provides a death benefit in exchange for premium payments. There are different types of life insurance, and term insurance, like Sara’s policy, provides coverage over a fixed period, after which the policy expires if the insured doesn’t pass away.

Insurers use statistical data and actuarial science to assess risks. This data includes the probability of death at different ages (as seen with Sara’s situation), which influences policy pricing. The challenge for insurance companies is to balance the premiums collected against the potential payouts while aiming for profitability.

• Sara’s policy is an example of term life insurance, focusing on coverage until age 65.
• Such policies are typically cheaper than whole life policies but offer no value if the insured survives the term.
• The calculated expected costs at each age help the insurer set a suitable premium that covers expected payouts and desired profit.

Statistical Analysis in Insurance

Statistical analysis in insurance involves using data to make predictions and decisions. It is essential in calculating expected costs and determining the premiums to charge. By analyzing the probability of events like death, insurers predict the likelihood of paying out a death benefit.

In Sara’s exercise, expected costs for each year from ages 60 to 64 were calculated, demonstrating how data drives decision-making in insurance. The total expected cost \( \text{2238.5} \) over her term is a sum of these yearly expected values, and it influences how much Sara will be charged for her policy.

Statistical tools help companies to:

• Estimate future liabilities based on historical rates of events.
• Set premium rates that cover these liabilities and produce profit.
• Continuously reassess and adjust predictions as new data becomes available.

Understanding statistical analysis ensures effective risk management, keeping both insurers and policyholders informed and prepared.