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Chapter 6: Problem 26

A life insurance company sells a \(\$ 250,000\) 1-year term life insurance policy to a 20-year-old male for \(\$ 350 .\) According to the National Vital Statistics Report, \(58(21),\) the probability that the male survives the year is \(0.998734 .\) Compute and interpret the expected value of this policy to the insurance company.

Short Answer

Expert verified
The expected value of the policy is approximately \( \$32.86 \). This means the insurance company expects to make about \( \$32.86 \) per policy on average.

Step by step solution

01

Identify the problem

Determine the expected value of the 1-year term life insurance policy sold to a 20-year-old male.
02

Define the variables

Let the insurance payout be \( P = \$250,000 \) and the premium paid be \( C = \$350 \). The probability that the male survives the year is \( P_{survive} = 0.998734 \) and the probability that he does not survive is \( P_{death} = 1 - P_{survive} = 1 - 0.998734 = 0.001266 \).
03

Calculate the Expected Value

The expected value is given by \( E(X) = P_{survive} \times \text{Profit}_{survive} + P_{death} \times \text{Profit}_{death} \), where \( \text{Profit}_{survive} \) is the profit when the male survives and \( \text{Profit}_{death} \) is the profit when he does not survive. If the male survives, the profit is simply the premium, \( \$350 \). If the male dies, the company pays out the coverage amount, so the profit is \( \$350 - \$250,000 = -\$249,650 \). Thus: \[ E(X) = 0.998734 \times 350 + 0.001266 \times (-249,650) \]
04

Simplify the calculation

Calculate each term separately: \[ 0.998734 \times 350 = 349.5569 \] \[ 0.001266 \times (-249,650) = -316.6969 \] Combine the results: \[ E(X) = 349.5569 - 316.6969 = 32.86 \]
05

Interpret the result

The expected value of the policy to the insurance company is approximately \( \$32.86 \). This means that on average, the insurance company expects to make \( \$32.86 \) from each policy it sells to 20-year-old males.

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Key Concepts

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

Probability
Probability is a way to measure how likely something is to happen. For example, when we say the probability that a 20-year-old male survives a year is 0.998734, we mean that there is a 99.8734% chance he will live through the year. This probability is calculated based on statistics and historical data. Understanding probability is essential in many areas, like insurance, because it helps companies predict and manage risks.
Expected Value
Expected value is a concept used to predict the average result of an event if we repeated it many times. In the context of insurance, it helps companies estimate their average profit or loss from policies. For instance, the expected value of the insurance policy for a 20-year-old male is calculated by weighing the outcomes by their probabilities. If the male survives, the company makes a profit of \(350, and if he doesn鈥檛, they incur a loss of \)249,650. The formula to calculate the expected value is: \text{Expected Value} = P_{survive} \times \text{Profit}_{survive} + P_{death} \times \text{Profit}_{death} This accounts for both the profits and losses adjusted by the probability of each.
Insurance Policy
An insurance policy is a contract between an insurer (the company) and an insured (the person). In the exercise, we look at a 1-year term life insurance policy. The insured pays a premium of \(350, and in return, the company agrees to pay \)250,000 if the insured dies within the year. The company uses calculated probabilities to decide the premium price, ensuring they, on average, make a profit by selling many policies.
Life Expectancy
Life expectancy refers to the average time a person is expected to live. It鈥檚 different for various groups based on age, gender, and health. In insurance, life expectancy is a crucial factor. For a 20-year-old male, the probability of surviving one year is very high, which means the chance of payout (if he dies) is low. Insurance companies utilize life expectancy data to set premiums. The more accurately they understand life expectancy, the better they can price their policies to manage risks and ensure profitability.

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A life insurance company sells a \(\$ 250,000\) 1-year term life insurance policy to a 20-year-old female for \$200. 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.

Historically, the probability that a passenger will miss a flight is 0.0995. Source: Passenger-Based Predictive Modeling of Airline No-show Rates by Richard D. Lawrence, Se June Hong, and Jacques Cherrier. Airlines do not like flights with empty seats, but it is also not desirable to have overbooked flights because passengers must be "bumped" from the flight. The Lockheed L49 Constellation has a seating capacity of 54 passengers. (a) If 56 tickets are sold, what is the probability 55 or 56 passengers show up for the flight resulting in an overbooked flight? (b) Suppose 60 tickets are sold, what is the probability a passenger will have to be "bumped"? (c) For a plane with seating capacity of 250 passengers, how many tickets may be sold to keep the probability of a passenger being "bumped" below \(1 \% ?\)
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