課程三 :Investment Decision with Certainty

• 課程重點 :– Introduce MIRR– Identify the relevant cash flows– An example– Discussing capital rationing– Hoemwork assignment– Excel tools

Problems with IRR

• Multiple IRRs

• Mutually exclusive projectsNPV method implicitly assumes that the rate at which cash flows can be reinvested is the cost of capital, whereas the IRR method implies that the firm has the opportunity to invest at the IRR.

Modified Internal Rate of Return (MIRR)• The discount rate at which the present value of a project’s c

ost is equal to the present value of its terminal value, where the terminal value is found as the sum of the future values of the cash inflows, compounded at the firm’s cost of capital.

• PV cost = PV terminal value

The future value of the cash inflows is also called the terminal value.

n

n

tn

n

t

tnt

tt

MIRR

TVtsPV

MIRR

kCIF

k

COF

)1(cos

)1(

)1(

)1(0

0

MIRR example

0 1 2 3 4

-1000 500 400 300 100

k=10% $330

k=10% $484

k=10% $665.5

TV 共 $1579.5PV 1000 MIRR=12.1%NPV=0

Advantages of MIRR

• MIRR assumes that cash flows from all projects are reinvested at the cost of capital.

• MIRR also solves the multiple IRR problem.• Mutually exclusive projects:

– Equal size, same life:same decision as NPV

– Equal size, different life: same

– Different size: could have different decision as NPV

Identify the relevant cash flows

• Only cash flow is relevant. Not accounting income

• Estimating cash flows on an incremental basis.– Do not confuse average with incremental payoffs.

– Include incidental effects. Externalities

– Do not forget working capital requirements.

– Forget sunk costs.

– Include opportunity costs.

– Be ware of allocated overhead costs.

• Be consistent in your treatment of inflation.

An example

1995 1996 1997 1998 1999Building (12000)Equipment (8000)Increase in NWC (6000)Sales 40000 40000 40000 40000Costs 29000 29000 29000 29000Depreciation(B) 156 312 312 312Depreciation(E) 1600 2560 1520 960Earning before taxes 9244 8128 9168 9728Taxes 3697.6 3251.2 3667.2 3891.2+Depreciation 1756 2872 1832 1272Cash flows 7302.4 7748.8 7332.8 7108.8Return of NWC 6000Net salvage value 10607Net cash flows (26000) 7302.4 7748.8 7332.8 23715.8NPV 6988

MACRS 1 2 3 4Building 1.3% 2.6% 2.6% 2.6%Equipment 20% 32% 19% 12%

Building EquipmentInitial cost 12000000 80000001999 salvage (market value) 7500000 20000001999 book vlaue 10908000 1360000Gain (loss) on sale (3408000) 640000Taxes -1363200 256000Net salvage value 8863200 1744000

1995 1996 1997 1998 1999-26000 7302 7749 7333 23716

• Payback period: 3.15 years.

• IRR: 21.9% versus a 12% cost of capital.

• MIRR:18.9 versus a 12% cost of capital.

• NPV:$6988

Effect of NWC on cash flow

0 1 2 3 4 5 6NWC 550 1289 3261 4890 3583 2002Cash flow due to NWC change -550 -739 -1972 -1629 1307 1581 2002

An example for treating inflation

• Assuming 15% nominal rate, and inflation is 10%.

Real Cash FlowsThousands of Dollars

C0 C1 C2 C3

-100 35 50 30

Two approaches

• Restate the cash flows in nominal terms and discount at the nominal rate.

• Restate the discount rate in real terms and use this to discount the real cash flows.

5.5)15.01(

1.1*30

)15.01(

1.1*50

)15.01(

1.1*35100

3

3

2

2

NPV

045.0110.1

15.11

rateinflation 1

ratediscount nominal1 Re

ratediscountal

5.5045.1

30

045.1

50

015.1

35100

32NPV

Comparing projects with unequal lives

• Replacement chain approach: assuming that each project can be repeated as many times as necessary to reach a common life span; the NPVs over this life span are then compared, and the project with the higher common life NPV is chosen.

0 1 2 3 4 5 6Project A -40000 8000 14000 13000 12000 11000 10000

NPV 6491 (k=12%)IRR 17.5

Project B -20000 7000 13000 12000NPV 5155 (k=12%)IRR 25.2

Span -20000 7000 13000 12000-20000 7000 13000 12000

NPV 8824 (k=12%)

Equivalent annual annuity (EAA) approach

1.Find each project’s NPV over its initial life.

2.There is a constant annuity cash flow that has the same present value as a project’s NPV.

3.The project with a higher EAA will always have a higher NPV when extended out to any common life.

0 1 2 3

EAA EAA EAA

PV1

PV2

PV3

NPV

Projects With Unequal Lives: Equivalent Annual Annuity

Year A B C 0 -$300,000 -$1,000,000 -$600,000 1 320,000 600,000 800,000 2 320,000 600,000 800,000 3 320,000 600,000 4 600,000 5 600,000 6 600,000

Find the value of an annuity that has the same life as each project:For Project A Solve the following for EAA:

NPV= $468,590 EAAA x PVIFA12%,3 = $468,590 you get EAAA = $195,100

For Project B Solve the following for EAA: NPV= $1,466,840 EAAB x PVIFA12%,6 = $1,466,840 you get EAAB = $356,774

For Project C Solve the following for EAA: NPV= $752,040 EAAC x PVIFA12%,2 = $752,040 you get EAAC = $444,968

Equivalent Annual CostsSame as Equivalent Annual Annuity Except with Costs

Consider the Following Example:

Assume machine A and B have cost payment schedule as follows,

which machine we should buy.Machine C0 C1 C2 C3 NPV(6%) A $15 $4 $4 $4 $25.69 B $10 $6 $6 $21.00

Again the technique involves finding the value of an annuity that has the same life as each machine.

For Project A: EAAA x PVIFA6%,3 = $25.69Equivalent Annuity Payment = $25.69/2.673 = $9.61

For Project B: EAAB x PVIFA6%,2 = $21.00Equivalent Annuity Payment = $21.00/1.833 = $11.45

Capital rationing

• Internal capital rationing(or soft rationing):

Managers either limit arbitrarily the total amount invested or the kind of investments the firm undertakes or set acceptance criteria that lead it to reject some investments that are advantageous when judged by market criteria.

• External capital rationing(or hard rationing)There is a difference between that market rate or interest which the firm can borrow money and the market rate at which it can lend.

cash flowproject c0 c1 c2 NPV PIA -10 30 5 21 3.1B -5 5 20 16 4.2C -5 5 15 12 3.4

Ranking by Profitability Index

• If budget limit is $10, we should accept project B and C.

More constraints

• A $10 budget limit applied to cash flows in each of year 0 and 1, which project we should choose, B & C, or A & D.

cash flowproject c0 c1 c2 NPV PIA -10 30 5 21 3.1B -5 5 20 16 4.2C -5 5 15 12 3.4D 0 -40 60 13 1.4

Linear Programming

Max 21 XA+16XB+12XC+13XD

s.t.

10 XA+5XB+5XC+0XD10

-30 XA-5XB-5XC+40XD10

0 XA 1

0 XB 1

0 XC 1

0 XD 1

First Homework Assignment

• Please use EXCEL to complete this assignment.• Questions:1. Using Payback Period Method,discounted payback period,

NPV, IRR, PI to decide whether accept individual projects.

2. Under each evaluation method, ranking projects. Describe any observation.

3. Assuming project 7 and 8 are mutually exclusive projects, using MIRR, EAA method to evaluate those projects.

discount rtate 0.120 1 2 3 4

-1300 350 500 450 550NPV $80.93

discount rtate0.1474170 1 2 3 4

-1300 350 500 450 550NPV $0.00

MAX 21 XA+16XB+12XC+13XD 46 10 XA+5XB+5XC+0XD L.E. 10 15 -30 XA-5XB-5XC+40XD L.E. 10 50 L.E. XA L.E. 1 10 L.E. XB L.E. 1 00 L.E. XC L.E. 1 10 L.E. XD L.E. 1 1

21 XA+16XB+12XC+13XD 34 10 XA+5XB+5XC+0XD L.E. 10 10 -30 XA-5XB-5XC+40XD L.E. 10 100 L.E. XA L.E. 1 10 L.E. XB L.E. 1 00 L.E. XC L.E. 1 00 L.E. XD L.E. 1 1