Question 2.1: As the owner of a landmark Chicago skyscraper, you decide to...
As the owner of a landmark Chicago skyscraper, you decide to purchase insurance that will pay $1 billion in the event the building is destroyed by terrorists. Suppose the likelihood of such a loss is 0.1%, the riskfree interest rate is 4%, and the expected return of the market is 10%. If the risk has a beta of zero, what is the actuarially fair insurance premium? What is the premium if the beta of terrorism insurance is -2.5?
Plan
The expected loss is 0.1% × $1 billion = $1 million.
Given a risk-free rate of 4% and an expected market return of 10%, we can use the Capital Asset Pricing Model (CAPM) to compute the rate of return we would use to compute the fair insurance premium under the two scenarios of a zero beta and a beta of -2.5. Once we have the rate of return, we will divide the expected loss (the cash flow) by 1 + the rate of return, as shown in Eq. 2.1.
Insurance Premium = \frac{Pr(Loss) \times E [ Payment in the Event of Loss ]}{1+r_{L}} (2.1)
Execute
If the risk has a beta of zero, we compute the insurance premium using the risk-free interest rate:
($1 million)/1.04 = $961,538
Given a beta for the loss, β_L , of -2.5, the required return is:
r_L=r_f+β_L(r_{mkt}-r_f)=4\%-2.5(10\%-4\%)=-11\%In this case, the actuarially fair premium is ($1 million)/(1 – 0.11) = $1.124 million.
Evaluate
Although this premium exceeds the expected loss when there is a negative beta, it is a fair price given the negative beta of the risk. The insurance pays off exactly when the cash flows from your business operations are likely to be very low.