Question

You can invest in a risk-free investment technology that requires an upfront payment of 1.07 million and will provide a perpetual annual cash flow of 118,000. Suppose all interest rates will be either 10.1% or 4.9% in one year and remain there forever. The risk-neutral probability that interest rates will drop to 4.9% to 92%. The one-year risk-free interest rate is 8.2% and today's rate on a risk-free perpetual bond is 5.1%. The rate on an equivalent bond that is repayable at any time (the callable annuity rate) is 9.1%.

a) What is the NPV of investing today

b) What is the NPV of waiting and investing tomorrow

c) Verify that the hurdle rate rule of thumb gives the correct time to invest in this case.

Answer 1

Solution:

a. Since the investment is risk-free, the cost of capital is the risk-free rate.

Using a hurdle rate equal to the callable annuity rate of 9.1%

NPV = $118,000/0.091 - $1,070,000

NPV = -$226,703

Since NPV today is negative, therefore you should not invest now.

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b.

When the NPV rate goes up 10.1%

NPV = 118,000/0.101 - 1,070,000

NPV = $98,316.83

When the NPV rate goes down 4.9%

NPV = 118,000/0.049 - 1,070,000

NPV = $1,338,163.27

The present value today of the expected NPV using risk-neutral probabilities is therefore

1,338,163.27 * 0.92 /1.082 = $1,137,809.80

c. Hurdle rule rate = $1,137,809.80

We should wait and invest a year later