Problem 22 - Algebra and Trigonometry

Question

Lightning Insurance An insurance company has determined that in a certain region the probability of lightning striking a house in a given year is about \(0.0003,\) and the aver- age cost of repairs of lightning damage is \(\$ 7500\) per incident. The company charges \(\$ 25\) per year for lightning insurance. (a) What is the company's expected value for the net income from each lightning insurance policy? (b) If the company has \(450,000\) lightning damage policies, what is the company's expected yearly income from lightning insurance?

Step-by-Step Solution

Answer

(a) Expected net income per policy is $22.75. (b) Total expected income is $10,237,500.

Step 1: Calculate Expected Loss Per Policy

To find the expected loss per policy, multiply the probability of lightning striking a house by the average cost of repairs. The formula to use is: \( \text{Expected Loss} = 0.0003 \times 7500 \).

Step 2: Calculate Expected Value of Net Income Per Policy

Subtract the expected loss per policy from the insurance premium to determine the expected net income per policy. The formula is: \( \text{Expected Net Income} = 25 - (0.0003 \times 7500) \).

Step 3: Calculate Total Expected Net Income from All Policies

Multiply the expected net income per policy by the total number of policies (450,000 policies) to find the company's expected yearly income. Use the formula: \( \text{Total Expected Income} = 450,000 \times (25 - (0.0003 \times 7500)) \).

Key Concepts

Expected Value

The concept of expected value is a key cornerstone in decision-making under uncertainty. In insurance, it helps companies predict potential financial outcomes over time. Expected value is essentially a weighted average of all possible outcomes. Each outcome is multiplied by the probability of it occurring. In the context of our problem, the probabilities of different outcomes — such as a lightning strike or no lightning strike — are used to calculate the financial impact of these events.

In this situation, the insurance company must determine the expected value of a lightning strike on a house. They do this by multiplying the probability of a lightning strike (0.0003) by the average repair cost (7500 USD). This gives the expected loss per policy due to lightning: \[ \text{Expected Loss} = 0.0003 \times 7500 \] By understanding the expected value of losses, the insurer can better set their policy prices to ensure profitability while remaining competitive.

Net Income Calculation

Calculating net income involves subtracting expected costs from total revenues. For the insurance company, this means assessing how much they earn from premiums versus how much they expect to pay out in claims.

In our example, the insurance company charges 25 USD per policy. To find the expected net income per policy, they subtract the expected loss (calculated earlier) from this premium:\[ \text{Expected Net Income} = 25 - (0.0003 \times 7500) \] This calculation provides insight into the average profit the company can anticipate per policy sold. Understanding net income calculation allows a company to make informed decisions about pricing and risk management. These calculations help assess whether the company is charging enough to cover potential losses while achieving desired profitability.

Insurance Policies

Insurance policies are contracts between the insurer and the insured that provide financial protection against losses. In the case of lightning insurance, the insurer agrees to cover repair costs if a house is struck by lightning.

Each policy generates income through the premiums charged. For the company in the problem, with 450,000 policies in force, calculating their total expected income involves multiplying the expected net income per policy by the number of policies:\[ \text{Total Expected Income} = 450,000 \times (25 - (0.0003 \times 7500)) \] Through risk pooling across many policies, insurance companies can protect themselves against individual policy losses by averaging out the costs. This risk management is central to their business model, allowing them to offer financial safeguard to homeowners while maintaining financial stability themselves.