Question

Fire! An insurance company estimates that it should make an annual profit of \(\$ 150\) on each homeowner's policy written, with a standard deviation of \(\$ 6000\). a) Why is the standard deviation so large? b) If it writes only two of these policies, what are the mean and standard deviation of the annual profit? c) If it writes 10,000 of these policies, what are the mean and standard deviation of the annual profit? d) Is the company likely to be profitable? Explain. e) What assumptions underlie your analysis? Can you think of circumstances under which those assumptions might be violated? Explain.

Short answer

Standard deviation is large due to claim unpredictability. For 10,000 policies: mean = $1,500,000, SD = $600,000. Likely profitable with assumptions of independence and identical distribution.

Step-by-step solution

Understanding Standard Deviation

The standard deviation is large because the profit on each policy can vary widely. This is due to the unpredictable nature of insurance claims, where some years might have high claims resulting in lower profits or even losses, while other years might have low claims leading to high profits.

Calculating Mean and Standard Deviation for 2 Policies

For 2 policies, the mean annual profit is calculated as: \( \text{Mean} = 2 \times 150 = 300 \). Since standard deviation of the sum of independent variables is the square root of the sum of their variances, the standard deviation follows: \( \text{SD} = \sqrt{2 \times 6000^2} = 6000 \sqrt{2} \approx 8485 \).

Calculating Mean and Standard Deviation for 10,000 Policies

For 10,000 policies, the mean annual profit is: \( \text{Mean} = 10,000 \times 150 = 1,500,000 \). The standard deviation is calculated as: \( \text{SD} = \sqrt{10,000 \times 6000^2} = 6000 \times 100 = 600,000 \).

Analyzing Profitability

The company is likely to be profitable because the large number of policies (10,000) means the law of large numbers applies, reducing relative variability. This makes it likely they will earn close to the expected mean profitability.

Examining Underlying Assumptions

The analysis assumes that profits from individual policies are independent and identically distributed. The assumptions might be violated if there are systemic risks affecting all policyholders, such as natural disasters, or if claim behavior is not independent.

Key concepts

Standard Deviation

The standard deviation is a critical measure in statistics that sheds light on the unpredictability or variability in a set of data points, like insurance profits in our case. It represents how much individual data points deviate from the mean, or average, value. For the insurance company in the exercise, the large standard deviation of $6000 signifies a high variability in profits for each policy. This can happen because insurance claims fluctuate greatly, depending on events like accidents or natural disasters.
Therefore, some years might result in high claims reducing profits, while others might have fewer claims, boosting profits substantially.

• A high standard deviation means there is a wide spread in possible outcomes, making profits unpredictable for a small number of policies.
• It also indicates high risk, which insurers must manage carefully.

Understanding these variabilities helps insurers to better model and prepare for different scenarios.

Law of Large Numbers

The law of large numbers is a fundamental concept in probability and statistics that states that as the number of trials increases, the sample mean will get closer to the expected value or mean of the population. In simpler terms, the more policies an insurance company writes, the closer their actual profit will be to the expected profit.
This problem illustrates well how the law of large numbers works in practice. If the insurance company writes only a few policies, their standard deviation is still quite high. But as they increase the number of policies to 10,000, the variability in terms of profits decreases relative to the expected over this large sample.

• This concept reassures insurers that with many policyholders, their mean profit will stabilize and reflect the expected average.
• However, it's crucial this principle only truly holds under the assumption that all policies are independent and identically distributed.

By writing a large number of policies, insurance companies aim to balance out small variations and risks, aligning more closely with predicted calculations.

Insurance Profit Analysis

Insurance profit analysis often involves careful scrutiny of possible gains and losses from underwriting policies. This process includes the understanding of concepts like mean and standard deviation, and statistical laws such as the law of large numbers to make sound financial forecasts.
In this exercise, the insurer aims to earn a certain profit annually across many policies. A crucial part of the analysis is determining whether these predictions are realistic given the inherent risks and variabilities.
When analyzing the potential profitability of writing 10,000 policies:

• The analysis shows a mean profit of $1,500,000, indicating a likely positive return.
• The standard deviation, though large initially, becomes relatively manageable in relation to the mean due to the extensive number of policies.

However, assumptions such as independence of policies and consistent claim behavior are vital to this analysis.
Potential threats include systemic risks like natural catastrophes that could lead to large-scale claims simultaneously affecting most policies.