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Choosing Between Mutually Exclusive Projects

Two Sticky Situations:

1. Choice between long- and short-lived projects: Should the firm save money today by investing in shorter-lived projects (such as less durable machinery)? ⇒ This is a job for Equivalent Annual Annuities 2. Investment timing and replacement decisions: ⇒ Should you invest in a new computer system today or invest in a new computer system next year? ⇒ When should existing machinery be replaced?

Equivalent Annual Annuities

Example: • You own a hotel in Bend. The location is killer but the place is run down. • Remodeling will cost you $100,000 but generate cash flows of $50,000 per year for the next five years. • Rebuilding will cost you $300,000 but generate cash flows of $66,000 per year for the next ten years. • Your opportunity cost of capital is 10% Cash Flows:

Time Remodel Rebuilding

0 1 2 3 4 5 6 7 8 9 10 -100 +50 +50 +50 +50 +50 -300 +66 +66 +66 +66 +66 +66 +66 +66 +66 +66

Equivalent Annual Annuities (cont.)

What is the NPV of remodeling your hotel?

Equivalent Annual Annuities (cont.)

What is the NPV of rebuilding your hotel?

50,000 50,000 50,000 50,000 50,000 NPV= −100,000 + + + + + 1.10 1.102 1.103 1.10 4 1.105 = 89,550 What is the IRR of remodeling your hotel?

NPV= −300,000 + = 105,570

66,000 66,000 66,000 66,000 + + + ...+ 1.10 1.10 2 1.103 1.1010

What is the IRR of rebuilding your hotel?

100,000 =

50,000 50,000 50,000 50,000 50,000 + + + + (1+ IRR) (1+ IRR) 2 (1+ IRR) 3 (1+ IRR) 4 (1+ IRR) 5 IRR = 40%

300,000 =

66,000 66,000 66,000 66,000 + + + ...+ (1+ IRR) (1+ IRR) 2 (1+ IRR) 3 (1+ IRR)10 IRR = 17%

Note: you could use Excel or a financial calculator to find IRR but I will not ask you to solve for it.

Equivalent Annual Annuities (cont.)

Summary: • Remodeling has an NPV of $89,550 and an IRR of 40% • Rebuilding has an NPV of $105,570 and an IRR of 17% • Therefore, IRR says remodel while NPV says rebuild. (Normally we would use NPV over IRR. But there is another way to analyze this situation.) What’s the problem? The projects have different lives! Time Remodel Rebuilding 0 1 2 3 4 5 6 7 8 9 10 -100 +50 +50 +50 +50 +50 -300 +66 +66 +66 +66 +66 +66 +66 +66 +66 +66

Equivalent Annual Annuities (cont.)

Definition: Equivalent Annual Annuity (EAA) converts the NPV of a project into annuity payments over the life of the project

1 1 NPV = EAA − r r ⋅ (1+ r) n

Note that this equation is equivalent to the PV annuity equation where NPV = PV and EAA = C Example: In the hotel example, EAA allows us to ask whether we want to (A) remodel the hotel twice or (B) rebuild it once… 1 1 89,550 NPV (A) = EAAA − = 23,614 ⇒ EAAA = .1 .1⋅ (1.1) 5 [10 − 6.21] 1 1 105,570 NPV (B) = EAAB − = 17,181 ⇒ EAAB = .1 .1⋅ (1.1)10 [10 − 3.855]

Are we doomed? No way! Equivalent Annual Annuity to the rescue!

1

6/27/2010

Equivalent Annual Annuities (cont.)

In other words, the annualized payout from rebuilding the hotel is $17,181 per year while the annualized payout from remodeling the hotel twice is $23,614 ⇒ We want to choose the project with the higher EAA since we are analyzing cash inflows ⇒ We want to remodel the hotel twice How do I know that $23,614 is the EAA of remodeling the hotel twice? First, what are the implied cash flows of remodeling the hotel twice? Time Remodel Rebuilding 0 1 2 3 4 5 6 7 8 9 10 -162 +50 +50 +50 +50 +50 +50 +50 +50 +50 +50 -300 +66 +66 +66 +66 +66 +66 +66 +66 +66 +66

Equivalent Annual Annuities (cont.)

Second, what is the NPV of the cash flows from remodeling twice (in thousands of dollars)?

NPV = −162 + or

50 50 50 + + ... + 1.10 1.102 1.1010

50 100 50 50 50 = −100 + + ... + + + ... + − 1.105 1.105 1.106 1.1010 1.10 = 145.15

where the present value of the remodeling COSTs in year zero and then again in year 5 is:

100,000 PV (Remodel twice) = −100,000 − = −162,092 1.10 5

So, remodeling your hotel...