Last Study Topics FCF and PV NPV and its competitors The payback Period The book rate of return Internal rate of return

Today Study Topics Internal Rate of Return Numerical

Internal Rate of Return The internal rate of return rule is to accept an investment project if the opportunity cost of capital is less than the internal rate of return; – If the opportunity cost of capital is less than the 28 percent IRR, then the project has a positive NPV. – If it is equal to the IRR, the project has a zero NPV. – And if it is greater than the IRR, the project has a negative NPV

Internal Rate of Return Pitfall 1 - Lending or Borrowing? Not all cash-flow streams have NPVs that decline as the discount rate 50% IRR = NPV = 0. Does this mean that they are equally attractive?

Internal Rate of Return Pitfall 1 - Lending or Borrowing? Not all cash-flow streams have NPVs that decline as the discount rate 50% IRR = NPV = 0. Does this mean that they are equally attractive? Project A = Lending Project B = Borrowing

Internal Rate of Return With some cash flows (as noted below) the NPV of the project increases as the discount rate increases. This is contrary to the normal relationship between NPV and discount rates. Discount Rate NPV

Project C If the opportunity cost of capital is 10 percent, that means the project is a good one. Or does it? Should we accept or reject? – The only way to find the answer is to look at the net present value.

Internal Rate of Return Pitfall 2 - Multiple Rates of Return Certain cash flows can generate NPV=0 at two different discount rates. The following cash flow generates NPV=0 at both (-50%) and (15.2%).

Internal Rate of Return 1000 NPV Discount Rate IRR=15.2% IRR=-50%

Internal Rate of Return Pitfall 3 - Mutually Exclusive Projects IRR sometimes ignores the magnitude of the project. The following two projects illustrate that problem. If you follow the IRR rule, you have the satisfaction of earning a 100 percent rate of return; If you follow the NPV rule, you are $11,818 richer;

Explanation Is it worth making the additional $10,000 investment in F? – you look at the IRR on the incremental flows.

Continue If the opportunity cost of capital were 20 percent, investors would place a higher value on the shorter-lived project G. Ignore Project I for the moment

Opportunity Cost matters But in our example the opportunity cost of capital is not 20 % but 10 %. Investors are prepared to pay relatively high prices for longer-lived securities, and so they will pay a relatively high price for the longer- lived project.

Internal Rate of Return

Continue At a 10 percent cost of capital, an investment in H has an NPV of $9,000 and an investment in G has an NPV of only $3,592; Assume, it is a shortage of capital which forces the choice between G and H; – But, When this implicit assumption is brought out, H is better if there is no capital shortage.

Alternative First, you check that project G has a satisfactory IRR. Then you look at the return on the additional investment in H. – The IRR on the incremental investment in H is 15.6 % > opportunity cost.

Internal Rate of Return Pitfall 4 - Term Structure Assumption We assume that discount rates are stable during the term of the project. This assumption implies that all funds are reinvested at the IRR. This is a false assumption. In a situation where it is important, we have to compare the project IRR with the expected IRR (yield to maturity).

Internal Rate of Return Expected IRR is the yield to maturity of the long term security that; – (1) is equivalent in risk to the project, – (2) offers the same time pattern of cash flows as the project. Calculating the IRR can be a laborious task. Fortunately, financial calculators can perform this function easily.

Summary Internal Rate of Return Pitfalls of IRR

Profitability Index (PI) When resources are limited, the profitability index (PI) provides a tool for selecting among various project combinations and alternatives A set of limited resources and projects can yield various combinations. The highest weighted average PI can indicate which projects to select.

Continue Example: The opportunity cost of capital is 10 percent, and our company has the following opportunities:

Continue We must pick the projects that offer the highest net present value (NPV) per dollar of initial outlay. This ratio is known as the profitability index. – PI=NPV / INV

Continue ProjectInvestment ($ Millions) NPV ($ Millions) PI A B C For our three projects the profitability index is calculated as follows;

Profitability Index Example We only have $300,000 to invest. Which do we select? ProjNPV InvestmentPI A230,000200, B141,250125, C194,250175, D162,000150,

Profitability Index Example - continued ProjNPV InvestmentPI A230,000200, B141,250125, C194,250175, D162,000150, Select projects with highest Weighted Avg PI WAPI (BD) = 1.13(125) (150) (25) (300) (300) (300) = 1.01

Profitability Index Example - continued ProjNPV InvestmentPI A230,000200, B141,250125, C194,250175, D162,000150, Select projects with highest Weighted Avg PI WAPI (BD) = 1.01 WAPI (A) = 0.77 WAPI (BC) = 1.12

Linear Programming Maximize Cash flows or NPV Minimize costs Example Max NPV = 21Xn + 16 Xb + 12 Xc + 13 Xd subject to 10Xa + 5Xb + 5Xc + 0Xd <= Xa - 5Xb - 5Xc + 40Xd <= 12

Numericals