Question

The Titanic Shipbuilding Company has a noncancelable contract to build a small cargo vessel. Construction involves a cash outlay of $268,000 at the end of each of the next two years. At the end of the third year the company will receive payment of $620,000. Assume the IRR of this option exceeds the cost of capital.

The company can speed up construction by working an extra shift. In this case there will be a cash outlay of $580,000 at the end of the first year followed by a cash payment of $620,000 at the end of the second year. Use the IRR rule to show the (approximate) range of opportunity costs of capital at which the company should work the extra shift. (Enter your answers as a percent rounded to 2 decimal places. Enter the smallest percent first.)

The company should work the extra shift if the cost of capital is between _______% and _____%.

I know the first blank is 22.82%, I just can't figure out the second one.

Answer 1

Cash flow for scenario 1:

Year 1: -268,000
Year 2: -268,000
Year 3: +620,000

Cash flow for Scenario 2:

Year 1: -580,000
Year 2: +620,000
Year 3: 0

Now, let us calculate differential of the cashflow -

Year 1: -580,000 - (-268,000) = -312,000
Year 2: 620,000 -(-268,000) = +888,000
Year 3: 0 - 620,000 = -620,000

Assume that IRR for differential cash flow is r.
Setup the equation for IRR -

0 = -312,000 + 888,000/(1+r) - 620,000/(1+r)²

let us assume (1+r) = x

so equation becomes, after replacing 1+r and cancelling zero's
312 - 888/x + 620/x² = 0

Multiply whole equation by x²
312x² - 888x + 620 = 0

Solve the quadratic equation -
ax² + bx + c = 0,

with two solutions -
x1 =
x2 =

in our case,
a = 312
b = -888
c = 620

so,
= = 121.5895
x1 = (888 - 121.5895)/(2*312) = 1.22822
x2 = (888 + 121.5895)/(2*312) = 1.61793

so,
1 + r = 1.22822 OR r = 22.82%
or 1 + r = 1.61793 OR r = 61.79%

So,

The company should work the extra shift if the cost of capital is between ___22.82____% and __61.79___%.

Important Note:
1. This is a classic case of multiple IRRs for a series of cashflow. When Cashflow changes sign more than once, multiple IRRs are possible. In this case sign of cashflow changed from negative to positive from 1st to 2nd year, then positive to negative again in 2nd to 3rd year. Because sign changed twice, we had a possibility of 2 IRRs.
2. It can be verified in excel that at cost of capital less than 22.82% and more than 61.79%, NPV of differential cashflow is negative, it is only positive when cost of capital is in between these values.
3. Curve between Cost of Capital and NPV looks like below in this case, where range of cost of capital for positive NPV can be seen -
(X-axis --> cost of capital)
(y-axis --> NPV)




Values in excel can be setup like below -


Values in excel look like below -


Screenshot shows value only till 27%, but formulas have been written till 80% of cost of capital.