91Ó°ÊÓ

Sara is a 60 -year-old Anglo female in reasonably good health. She wants to take out a 50,000 dollar term (i.e., 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). $$\begin{array}{|l|lcccc|} \hline x=\text { age } & 60 & 61 & 62 & 63 & 64 \\\ \hline P( \text { death at this age) } & 0.00756 & 0.00825 & 0.00896 & 0.00965 & 0.01035 \\ \hline \end{array}$$ 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 dollar 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) If Big Rock Insurance wants to make a profit of 700 dollar above the expected total cost paid out for Sara's death, how much should it charge for the policy? (d) If Big Rock Insurance Company charges 5000 dollar for the policy, how much profit does the company expect to make?

Step by step solution

01

Calculate Probability for Year 60

The probability that Sara will die in her 60th year is given by the Vital Statistics table as 0.00756. This means there is a 0.756% chance of her dying during this period.

02

Expected Cost for Year 60

The expected cost for the insurance company is calculated by multiplying the probability of death by the death benefit: \( 0.00756 \times 50,000 = 378 \) dollars.

03

Calculate Probability for Year 61

The probability that Sara will die in her 61st year is 0.00825. This translates to a 0.825% chance of her dying during this period.

04

Expected Cost for Year 61

Similarly, the expected cost for the insurance company for her 61st year is \( 0.00825 \times 50,000 = 412.5 \) dollars.

05

Calculate Probability for Year 62

The probability that Sara will die in her 62nd year is 0.00896, equivalent to a 0.896% chance.

06

Expected Cost for Year 62

For year 62, the expected cost is \( 0.00896 \times 50,000 = 448 \) dollars.

07

Calculate Probability for Year 63

The probability of death in the 63rd year is 0.00965, which means a 0.965% chance.

08

Expected Cost for Year 63

For the 63rd year, the expected cost is \( 0.00965 \times 50,000 = 482.5 \) dollars.

09

Calculate Probability for Year 64

The probability that Sara will die in her 64th year is 0.01035, translating to a 1.035% chance.

10

Expected Cost for Year 64

For the 64th year, the expected cost to the insurance is \( 0.01035 \times 50,000 = 517.5 \) dollars.

11

Total Expected Cost To Insurance

To find the total expected cost over 5 years (60-64), we sum the expected costs for each year: \( 378 + 412.5 + 448 + 482.5 + 517.5 = 2238.5 \) dollars.

12

Calculate Insurance Premium for $700 Profit

The policy charge needed to ensure a $700 profit over the expected costs is \( 2238.5 + 700 = 2938.5 \) dollars.

13

Expected Profit for $5000 Policy

If the company charges $5000 for the policy, the expected profit is \( 5000 - 2238.5 = 2761.5 \) dollars.

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Key Concepts

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

Life Insurance

Life insurance is an agreement between an individual and an insurance company, where the company provides a payout, known as a death benefit, to the beneficiaries when the insured person passes away. In the context of term life insurance, which is often referred to simply as term insurance, the policy only offers coverage for a specific period or term. The policy usually does not accumulate cash value and is intended solely to provide financial protection during the term. In Sara's case, she opts for a term life insurance policy that is active until the age of 65, with a death benefit of $50,000. This means if she unfortunately passes away before her 65th birthday, the specified amount will be paid to her beneficiaries.

Expected Cost

Expected cost in the context of life insurance involves calculating the anticipated payout the insurer must prepare for in the event of the insured person's death during each year of the term. For each year of the policy, the expected cost is determined by multiplying the probability of death in that year by the death benefit amount. For example, in Sara's 60th year, it’s calculated as:

• Probability of death: 0.00756 or 0.756%
• Death benefit: $50,000

The expected cost is \[0.00756 \times 50,000 = 378 \text{ dollars}\]This value represents what the company anticipates potentially paying out in that year based on statistical probabilities. Such calculations are performed for every year the policy is active, allowing the insurance company to adequately estimate its financial obligations.

Probability Calculation

Probability calculation is a core element in determining the risk factor associated with life insurance policies. This involves analyzing statistical data, such as the probability of death, to make informed decisions about policy pricing and risk assessments. In Sara’s situation, the provided probabilities for each year reflect the likelihood of her death at specific ages:

• Age 60: 0.00756 or 0.756%
• Age 61: 0.00825 or 0.825%
• Age 62: 0.00896 or 0.896%
• Age 63: 0.00965 or 0.965%
• Age 64: 0.01035 or 1.035%

Each probability is a crucial input in the expected cost calculations, directly influencing the financial strategies of insurance companies.

Profit Calculation

Profit calculation for an insurance company involves determining the difference between the revenue from policy premiums and the expected costs over the policy term. Insurance companies aim to balance offering competitive pricing with ensuring profitability. For Sara’s policy, Big Rock Insurance targets a profit margin. By summing the expected costs across all years, they determined the total expected cost to be \(2238.5. To realize a profit, a premium is set higher than this total expected cost. If the desired profit margin is \)700, the necessary premium is:\[2238.5 + 700 = 2938.5 \text{ dollars}\]Should they charge $5000, as considered, the profit expected would significantly increase, yielding:\[5000 - 2238.5 = 2761.5 \text{ dollars}\]This illustrates how insurance companies use math to both manage risk and maintain business sustainability.