Lending or borrowing position

 

If a project offers positive cash flows followed by negative flows, NPV can rise as the discounted rate is increased. Manager should accept such projects if their IRR is less than the opportunity cost of capital. Not all cash flow streams have NPVs that decline as the discount rate increases. Consider the following projects:

 

  Cash Flows ($)
Project C0 C1 IRR (%) NPV at 10%
A -1.000 +1.500 +50 +364
B +1.000 -1.500 +50 -364

 

Table 3.5

Each project has an IRR of 50%. Does this mean that project A and B are equally attractive? It is clear not, for in the case of A, where is the manager initially paying out $1,000, and is lending money at 50%; in the case B, where is the manager initially receiving$1,000, and is borrowing money at 50%. When manager lend money, manager want high rate of return; when manager borrow money, manager want a low rate of return. Obviously the internal rate of return rule won’t work in this case; manager has to look for an IRR less than the opportunity cost of capital.

Multiple rates of returns

If there is more than one change in the sign of the flows, the project may have several IRRs or no IRR at all. Suppose the project A involves an initial investment of $60 million and expected to produce a cash inflow of $12 million a year for the next nine years. At the end of that time the company will incur $15 million of cleanup cost.

 

Cash flow
C0 C1 C9 C10
-60   -15

Table 3.6

 

The project’s IRR and its NPV as follows:

 

IRR (%) NPV at 10%

-44.0 and 11.6 $3.3 million

 

There are two discounted rates that makes NPV=0

 

NPV =

 

NPV =

 

In other words, the investment has an IRR of both -44.0 and 11.6 percent. It happened because the double change in the sign of the cash flow stream. There can be as many internal rates of return for the project as there are changes in the sign of the cash flow ( By Descartes’ “rule of sign” there can be as many different solution to a polynomial as there are changes sign).

 

 

 

 


IRR= - 44%; IRR=11.6%; NPV=0. Figure 3.2

 

Figure show how this comes about. As the discount rate increases, NPV initially rises and then declines. There are also cases in which no internal rate of return. For example, project has a positive NPV at all discount rate:

 

  Cash Flows ($)
Project C0 C1 C2 IRR NPV at 10%
  +1.000 -3.000 +2.500 None +339

 

Table 3.7

 

A number of adaptations of the internal rate of return have been devised for such cases. Not only are they inadequate, but they also unnecessary for the simple solution is to use NPV. Companies sometimes get around the problem of multiple rates of return by discounting the later cash flow back at the cost of capital until there reminds only one change in the sign of the cash flows. A Modified Internal Rate of Return (MIRR) can than be calculated on this revised series. In our first example the MIRR is calculate as follows:

 

1. Calculate the PV of the year 9 and 10 cash flow:

PV in year

2. Add to the year 8 cash flow the PV of subsequent cash flows:

 

C8 + PV (subsequent cash flows) = +12 – 1.49 = 10.51

 

3. Since there is now only one change in the sign of the cash flows, the revised series has a unique rate of return, which is 11.5 %:

NPV =

 

Since the MIRR of 11.5 % is greater than the cost of capital, the project has positive NPV when valued at the cost of capital. Of course, it would be much easier in such case to abandon the IRR rule and just calculate project NPV.