Chapter 30, Problem 1P

The William Companies (WMB) owns and operates natural gas pipelines that deliver 12% of the natural gas consumed in the United States. WMB is concerned that a major hurricane could disrupt its Gulfstream pipeline, which runs 691 miles through the Gulf of Mexico. In the event of a disruption, the firm anticipates a loss of profits of $65 million. Suppose the likelihood of a disruption is 3% per year, and the beta associated with such a loss is –0.25. If the risk-free interest rate is 5% and the expected return of the market is 10%, what is the actuarially fair insurance premium?

Summary Introduction

To determine: Insurance premium.

Introduction:

To manage risk, a firm pays a yearly payment called insurance premium, instead of purchasing the insurance to pay off the losses.

Answer to Problem 1P

The insurance premium is $1.88 million.

Explanation of Solution

Given information:

Loss of profit is $65 million, disruption is 3%, beta is -0.25, risk-free interest rate is 5%, and the expected return is 10%.

The formula to calculate the insurance premium:

$\text{Insurance premium}=\frac{\text{P}\left(\text{Loss}\right)\times \text{E}\left(\text{Payment in the Event of Loss}\right)}{{\text{l + r}}_L}$

Here,

P refers to the probability.

E refers to the expected payment of loss.

${\text{r}}_L$ refers to the appropriate cost of capital.

The formula to calculate the cost of capital:

$\text{Cost of capital}=\text{Risk-free interest rate}-\text{Beta}\left(\text{Expected return}-\text{Risk-free interest rate}\right)$

Compute the appropriate cost of capital:

$\begin{array}{c}\text{Cost of capital}=\text{Risk-free interest rate}-\text{Beta}\left(\text{Expected return}-\text{Risk-free interest rate}\right)\\ =0.05-0.25\left(0.10-0.05\right)\\ =0.05-0.0125\\ =0.0375\end{array}$

Hence, the cost of capital is 0.0375.

Compute the insurance premium:

$\begin{array}{c}\text{Insurance premium}=\frac{\text{P}\left(\text{Loss}\right)\times \text{E}\left(\text{Payment in the Event of Loss}\right)}{{\text{l + r}}_L}\\ =\frac{0.03\times \$65,000,000}{1+0.0375}\\ =\$1.88\text{million}\end{array}$

Hence, the insurance premium is $1.88 million.