Suppose That A Car Rental Agency Offers Insurance For A Week

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Suppose that a car rental agency offers insurance for a week that will cost $10 per day. A minor fender bender will cost $1,500, while a major accident might cost $15,000 in repairs. Without the insurance, you would be personally liable for any damages. What should you do? Clearly, there are two decision alternatives: take the insurance or do not take the insurance.

The uncertain consequences, or events that might occur, are that you would not be involved in an accident, that you would be involved in a fender bender, or that you would be involved in a major accident. Assume that you researched insurance industry statistics and found out that the probability of major accident is 0.05%, and that the probability of a fender bender is 0.16%. What is the expected value decision? Would you choose this? Why or why not?

What would be some alternate ways to evaluate risk?

Paper For Above instruction

When renting a vehicle, the decision to purchase insurance involves careful risk assessment and financial analysis. By examining the probabilistic outcomes and their associated costs, one can determine whether purchasing the insurance policy provides a financial advantage. This paper explores the expected value approach to evaluating insurance decisions, calculates the expected costs with and without insurance, and discusses alternative risk evaluation methods.

Introduction

Car rental insurance decisions often pose a dilemma for consumers: is it worth paying a daily fee to mitigate potential financial losses resulting from vehicle damages? The total cost of damages varies based on severity, with minor fender benders costing significantly less than major accidents. Probabilistic data from industry statistics inform us of the likelihood of different accident types, enabling an expected value analysis to guide decision-making.

Expected Value Calculation

To determine whether to purchase insurance, we need to calculate the expected costs associated with each option—buying the insurance versus not buying it. The cost of the insurance is straightforward: $10 per day over a week, totaling $70. Without insurance, costs are contingent on accident probabilities:

• No accident occurs: Probability = 1 - (probability of fender bender + probability of major accident) = 1 - (0.0016 + 0.0005) = 0.9979.
• Fender bender: Probability = 0.0016, cost = $1,500.
• Major accident: Probability = 0.0005, cost = $15,000.

Calculating expected damages without insurance:

Expected Damage = (Probability of Fender Bender × Cost) + (Probability of Major Accident × Cost)

= (0.0016 × $1,500) + (0.0005 × $15,000)

= $2.40 + $7.50 = $9.90

Therefore, the expected cost of damages without insurance is approximately $9.90.

Decision Analysis

Adding the insurance fee of $70, the total expected cost if insurance is purchased is:

Total Cost with insurance = Insurance cost + Expected damages = $70 + $9.90 = $79.90.

Without insurance, the expected cost is only the expected damages, approximately $9.90. Even considering the insurance premium, the total expected expenditure ($79.90) is significantly higher than the expected damages without insurance ($9.90), suggesting that purchasing insurance, based solely on expected monetary value, is not economically justified in this scenario.

Interpretation and Decision

From a purely quantitative standpoint, the expected value analysis indicates that not purchasing insurance minimizes expected costs. However, individual risk tolerance and psychological comfort with potential large out-of-pocket expenses might influence the decision. For risk-averse individuals, purchasing insurance may provide peace of mind despite the higher expected monetary cost. Conversely, risk-tolerant individuals might prefer to gamble on the low probability of a major accident and save the insurance costs.

Alternative Ways to Evaluate Risk

While expected value provides a valuable quantitative framework, other methods can enhance risk evaluation. These include:

• Utility Theory: Considers an individual's subjective value or utility of outcomes, acknowledging that the monetary impact of damages might be perceived differently based on personal risk preferences.
• Scenario Analysis: Examines possible outcomes under various scenarios, including worst-case situations, to assess potential risks beyond average expected losses.
• Monte Carlo Simulations: Uses computational algorithms to simulate numerous possible outcomes to understand the distribution and likelihood of different results more comprehensively.
• Risk-Tolerance Assessment: Incorporates personal risk thresholds and financial resilience levels into decision-making, moving beyond purely probabilistic calculations.
• Cost-Benefit Analysis: Compares the tangible benefits of insurance (peace of mind, financial protection) against its costs to evaluate overall value.

Conclusion

The decision to purchase rental car insurance hinges on both quantitative analysis and personal risk preferences. The expected value calculation suggests that, from a purely financial standpoint, not buying insurance minimizes expected costs. However, personal risk aversion, potential emotional distress, and broader risk evaluation strategies might lead individuals to favor insurance despite its higher expected monetary cost. Incorporating alternative risk assessment methods ensures a more holistic decision-making process aligned with individual circumstances and preferences.

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