Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A man wishes to purchase a 5 -yr term-life insurance policy that will pay the beneficiary $$\$ 20,000$$ in the event that the man's death occurs during the next 5 yr. Using life insurance tables, he determines that the probability that he will live another \(5 \mathrm{yr}\) is \(.96\). What is the minimum amount that he can expect to pay for his premium?

Step-by-Step Solution

Answer

The minimum amount the man can expect to pay for his premium is $800.

Step 1: Calculate the probability of dying within the next 5 years

Since the probability of the man living another 5 years is 0.96, we can find the complementary probability of him dying within the next 5 years by subtracting this from 1: Probability of dying = \(1 - 0.96 = 0.04\)

Step 2: Calculate the minimum premium amount

Since the policy will pay the beneficiary \(20,000 if the man dies within the next 5 years, the insurance company will make this payment with a probability of 0.04. To cover their potential losses, the company will collect insurance premiums from the man. The minimum amount the man can expect to pay for his premium can be found by multiplying the payout (\)20,000) by the probability of dying (0.04): Minimum premium amount = \(20000 \times 0.04 = 800\) So, the minimum amount that the man can expect to pay for his premium is $800.

Key Concepts

Term-Life Insurance Policy

When discussing life insurance, a 'term-life insurance policy' is a pivotal concept that deserves attention. In essence, it is a type of life insurance that provides coverage at a fixed rate of payments for a limited period, the relevant 'term'. After that period expires, coverage is no longer guaranteed, and the client must either forgo coverage or potentially obtain further coverage with different payments or conditions.

These policies are significant in financial planning, particularly for individuals seeking peace of mind in knowing they can provide for their beneficiaries in the event of their untimely death. This type of policy is succinctly defined by its simplicity and typically lower initial costs compared to permanent life insurance.

Probability Calculation

The 'probability calculation' is a fundamental aspect of both life insurance models and broader statistical analysis. Probability, mathematically represented as a number between 0 and 1, quantifies the likelihood of an event. When an event is certain not to happen, the probability is 0, and when it is certain to occur, the probability is 1.

To calculate the probability of an individual event within a certain time frame (like death within 5 years for life insurance), actuaries use life tables that provide statistical probabilities based on factors such as age, gender, and sometimes even lifestyle choices or health status. This numerical calculation informs critical decision-making, such as premium setting in the context of life insurance.

Complementary Probability

The notion of 'complementary probability' addresses the probability of the occurrence of the opposite (complementary) event. For every event there exists another event that is the opposite, and the sum of the probabilities of these two events must equal 1.

For example, if the probability of living for 5 more years is 0.96, then the complementary probability, or the probability of dying within those 5 years, is calculable as 1 - 0.96, which equals 0.04. This concept is especially useful in life insurance calculations, where insurers need to evaluate both the likelihood of a payout event occurring and its opposite to gauge risks.

Expected Value

In the realm of statistics and probability, 'expected value' is a concept that refers to the anticipated value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence. It represents the average amount one can expect to be paid out or to pay in a large number of similar scenarios.

In life insurance, the expected value of a policy to the company is the payout amount multiplied by the probability of the event (the insured person's death) happening within the insured term. This figure helps insurance companies to determine the minimum premium that they should charge to ensure that they will, on average, cover the payouts they are obliged to make, thus maintaining profitability.