Question 2.3: Your firm faces a potential $100 million loss that it would ...
Your firm faces a potential $100 million loss that it would like to insure. Because of tax benefits and the avoidance of financial distress and issuance costs, each $1 received in the event of a loss is worth $1.50 to the firm. Two policies are available: One pays $55 million and the other pays $100 million if a loss occurs. The insurance company charges 20% more than the actuarially fair premium to cover administrative expenses. To account for adverse selection, the insurance company estimates a 5% probability of loss for the $55 million policy and a 6% probability of loss for the $100 million policy.
Suppose the beta of the risk is zero and the risk-free rate is 5%. Which policy should the firm choose if its risk of loss is 5%? Which should it choose if its risk of loss is 6%?
Plan
The premium for each policy will be based on the expected loss using the insurance company’s estimate of the probability of loss:
$55 million policy: 5% chance of loss $100 million policy: 6% chance of loss
Because it is charging 20% more than the actuarially fair premium, the insurance company will set the premium at 1.20 times the present value of expected losses.
However, the value of the policy to you depends on your estimate of the true probability of loss, which is based on your own assessment and does not depend on the size of the policy. Because each $1 of insured loss benefits your firm by $1.50, you are willing to pay 1.50 times the present value of the expected loss.
Because the beta of the risk is 0, the risk-free rate of 5% is the appropriate discount rate for all calculations.
Execute
The premium charged for each policy is:
Premium (\$55 million policy) =\frac{5\%\times \$55 million}{1.05} \times 1.20=\$3.14 million Premium (\$100 million policy) =\frac{6\%\times \$100 million}{1.05} \times 1.20=\$6.86 millionIf the true risk of a loss is 5%, the NPV of each policy is:
NPV ($55 million policy):
=-\$3.14 million+\frac{5\%\times \$55 million}{1.05} \times 1.50=\$0.79 millionNPV ($100 million policy):
=-\$6.86 million+\frac{5\%\times \$100 million}{1.05} \times 1.50=\$0.28 millionThus, with a 5% risk, the firm should choose the policy with lower coverage. If the risk of a loss is 6%, the policy with higher coverage is superior:
NPV ($55 million policy):
=-\$3.14 million+\frac{6\%\times \$55 million}{1.05} \times 1.50=\$1.57 millionNPV ($100 million policy):
=-\$6.86 million+\frac{6\%\times \$100 million}{1.05} \times 1.50=\$1.71 millionEvaluate
Note that the insurance company’s concerns regarding adverse selection are justified: Firms that are riskier will choose the higher-coverage policy.