Question

You can invest in a​ risk-free technology that requires an upfront payment of $1.13

million and will provide a perpetual annual cash flow of $115,000.

Suppose all interest rates will be either 9.7% or 4.7% in one year and remain there forever. The​ risk-neutral probability that interest rates will drop to 4.7% is 93%. The​ one-year risk-free interest rate is 8.1%​,

and​ today's rate on a​ risk-free perpetual bond is 5.5%. The rate on an equivalent perpetual bond that is repayable at any time​ (the callable annuity​rate) is 8.6%.

What is the NPV of investing​ today?

What is the NPV of waiting and investing​ tomorrow? The NPV if the rate goes up is $? The NPV if the rate goes down is $? The PV is?

Verify that the hurdle rate rule of thumb gives the correct time to invest in this case. The hurdle rule is $?

The NPV is >0 so should you invest now or wait?

Answer 1

What is the NPV of investing​ today?

Here the appropriate cost of capital to be used for capitalization of perpetual cash flows is todays's rate on a risk free perpetual bond = 5.5 %

NPV = Annual CIF / r - Upfornt Investment = 115,000 / (0.055) - 1130,000 = 960,909

What is the NPV of waiting and investing​ tomorrow?

Here we have to use discount rate in computing the present value = The​ one-year risk-free interest rate i= 8.1%​, For capitalizing the cash flows, we have use the rates as they were after one year ( i.e either 9.7 % or 4.7% )

The NPV if the rate goes up is $?

NPV _{rate go up} = 115,000 / (0.097) - 1130,000 = 55567

The NPV if the rate goes down is $? The PV is?

NPV _{rate goes down} = 115,000 / (0.047) - 1130,000 = 1316809

PV = Present value = [ 55567 * (0.07) + 1316809 * (0.93) ] / (1.081)

= [ 3889.69 + 1224632 ] / (1.081) = 1,136,467.80

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

Here we have apply r = callable bond annuity rate = 8.6% for computing the NPV

NPV = 115000 / 0.086 - 1130,000 = 207,209.30

The hurdle rule is $? ..........   = 207,209.30 The NPV > 0 so should Invest now

Explanation. Hurdle rate method shall provide only an approximation. As seen in the above calulations the present value to invest after one year is 1,136,467.80 as against investing now of 960,909. Thus we should wait and invest after one year, but hurdle rate suggest immediate investment.