Question

Acme Inc. owns a delivery truck worth $95,000. The truck is subject to the risk of physical damage, with the following loss probability distribution:

Annual Losses	Probability
$0	.85
$10,000	.10
$95,000	.05

Acme’s risk manager is considering three risk management options to manage this risk:

• Acme can purchase full insurance on this truck for the risk of physical damage up to a total loss of $95,000 for a premium of $3,500.
• Acme can purchase insurance with a limit of $95,000 and a deductible of $1,000 for a premium of $2,200.
• Acme is also considering retention as an alternative to full insurance.

1. Construct a loss matrix. All matrices must be constructed with the Risk Management Alternatives as the rows and the States of the World as the columns, as we did in the video. Ignore tax considerations for this assignment.

2. Suppose the risk manager wants to minimize expected loss as her decision rule. Which risk management alternative does she choose? Show all expected loss calculations and explain why the risk manager chooses a particular option.

3. Assume that the risk manager has a worry value (WV) equal to $1,500 for retention and a WV of $1,200 for deductible insurance. If the risk manager decides to minimize TOTAL COST, which risk management alternative does she choose? Show all total cost calculations and explain why the risk manager selects a particular option.

Answer 1

Annual Losses	Probability	Prob weighted Loss
$0	0.85	$0
$10,000	0.1	$1,000
$95,000	0.05	$4,750
Total	1	$5,750
State of World	no loss	$10,000 loss	$95,000 (100%) loss
Risk mgmt Alternative	Payoff			Prob Weighted Loss
Full insurance	-3500	6500	91500	2250
Insurance with deduct	-2200	6800	91800	3400
Retention	0	-10000	-95000	-5750
Probability	0.85	0.1	0.05

As seen from the above loss matrix, if the risk manager wants to reduce the expected loss, she can choose the insurance option with $1000 deductible. It has the lowest probability weighted/ expected loss of 3400. Note- +ve loss value is actually a gain. Hence, an alternative with +ve value in the above loss matrix has a net positive payoff based on probabilities. For insurance with deductible, the probability weighted benefit of lower premium outweighs the probability weighted cost of deductible. Hence, she should chose this option.

The WV is the cost of alternative. The total loss would comprise of the expected loss plus the worry value. We can directly add the WV to the probability weighted loss obtained above.

Risk mgmt Alternative	Prob Weighted Loss	Worry Value	Total cost
Full insurance	2250	0	2250
Insurance with deduct	3400	-1200	2200
Retention	-5750	-1500	-7250

As seen above, the total cost is lowest for full insurance once we include the worry value. Note: +ve loss value is actually a gain. Hence, an alternative with +ve value in the above loss matrix has a net positive payoff based on probabilities. Once we include the cost of worry, the last 2 alternatives become more costly. Hence, full insurance becomes the best alternative based on full cost consideration.