TIME VALUE OF MONEY• Compounding & Discounting

– Earlier value…later value– Implicit rate– Implicit time

• Annuities– PVA--Credit card problem– FVA--Sinking fund problem

• Effective rates– EAR and APR– The “RIPOV FURNITURE” case

Generalization

• Formula

• with factor

mn

m

iPVFV )1(

m

imn

FVIFPVFV;

Example

• Assume that we can find an investment that returns 20% per annum with semi-annual compounding. How much should we invest today in order to become a millionaire ten years from now?

Solution• Method 1: “Crunch numbers”

– PV=FV/(1+i/m)n x m =1,000,000/(1+.2/2)20

– PV=148,643.6

• Method 2: use the table– PV=FV / FVIF(i/m; n x m)=1000,000/ FVIF(10%;20)

– PV= 1,000,000 /6.7275 =148,643.6

• Method 3: Use the calculator– FV=1,000,000; I=10%; n=20; COMPUTE PV

Implicit Rate

i Computen PV- FV :

or

or

1

);(

1

Input

PV

FVFVIF

PV

FVmi

m

imn

mn

Example

• Suppose an investment offers to triple your money in 2 years (don’t believe it). What rate are you being offered, if interest rates are compounded twice a year?

Solution

%642 31.9%iThen

%9.31i i CPTn 24 PV 1- FV 3

%64%322

3

%6411

321

)2

;4(

4

11

ii

FVIF

PV

FVmi

i

mn

Implicit time

n Compute i PV- FV :

or

or

)1ln(

)ln(1

);(

Input

PV

FVFVIF

mi

PVFV

mn

m

imn

Example

• You have $430 today. You need $500. If you earn 1 percent per month, how many months will you wait?

Solution

months 15nThen

n Compute i 1 PV 430- FV 500

or

1512163.1430

500

or

1526.1)

12%12

1ln(

)430500

ln(

12

1

)1ln(

)ln(1

%)1;12(

monthsnFVIF

months

mi

PVFV

mn

n

Ordinary Annuities

• Present value of an annuity: Credit card payments, mortgage payments…– PAY-OFF A DEBT

• Future value of an annuity: Sinking funds, annuity funds, insurance premium…– INVEST TO ACCUMULATE

PV of an annuity:

• Formula:

• With factor:

mimi

PMTPVA

mn)1(

11

m

imn

PVIFAPMTPVA;

EXAMPLE

• Suppose that you have maxed out your credit card at $5,000. The APR on the card is 24%; you make monthly payments and interests are compounded monthly. What payment should you make if you want to reimburse everything in 2 years?

Answer

• COMPUTATION:– PVA=PMT x PVIFA(n x m; i/m)

– PMT=PVA/PVIFA(n x m; i/m)

– =5000/PVIFA(24; 2%)=5000/18.9139=$264.36

• CALCULATOR:– PV=5000; I=2%; n=24; COMPUTE PMT

FV of an annuity

• Formula:

• with factor:

mi

mi

PMTFVA

mn 1)1(

m

imn

FVIFAPMTFVA;

EXAMPLE

• How much should you invest each six months in order to receive $1,000,000 in ten years in an investment that is expected to return 20%?

Answer

• COMPUTATION:– FVA=PMT x FVIFA(m x n; i/m)

– PMT=FVA/ FVIFA(m x n; i/m)

– PMT=1,000,000/FVIFA(20;10%)

– PMT=1,000,000/57.275=$17,459.6

• CALCULATOR:– FV=1000000; I=10%; n=20 COMPUTE PMT

APR Vs EAR

• EAR is the true rate: it includes the compounding effect

• EAR=(1+APR/m)m-1

• EXAMPLE: for the monthly payments on the 24% credit card, the EAR is:– EAR=(1+i/m)m-1=(1+24%/12)12-1– EAR=1.2682-1=26.82%

EXCEL Functions

• InsertFunctionFinancial• PV (i/m , nxm , PMT , FV , Type)Calculate PV or PVA• FV (i/m , nxm , PMT , PV , Type)Calculate FV or FVA• PMT (i/m , nxm , PV , FV , Type)Calculate PMT• Rate (nxm , PMT , PV , FV , Type, Guess)Calculate i/m• NPER (i/m , PMT , PV , FV , Type)Calculate n x m

Capital Budgeting• Real Asset Valuation and Profitability

– NPV– IRR– MIRR– Payback– Cross-over rates

• Capital Budgeting process– Cash flows that matter– WACC– Sensitivity analysis– Incorporating risk in capital budgeting

Real Asset Valuation

• Valuation• Measuring Profitability

– The good (NPV)– The bad (IRR)– Cross-over rate– The ugly (Payback)– MIRR

Real Asset Valuation• PV(asset)=PV(future cash flows from asset)

• 3 elements:– CF=cash flow

– Maturity=n

– Interest rate=RAverage cost of moneyCost of capital?

• What are the determinants of the firm’s value?• What would the firm’s value be if it had a perpetual cash

flow?• Can the firm get value from other factors?

nRCFn

RCF

RCF

RCF

assetPV)1(

...)1(

3)1(

2)1(

1)(

32

The good: Net Present Value

• Formula:

• Where I/O is the initial outlay• It measures the $ profitability, taking into account

time value of money and risk. It is often referred to as the “extra” $ available to the owners…any comments?

• It assumes that cash flows are reinvested at R.

OIR

CFnR

CFR

CFR

CFassetNPV

n/

)1(...

)1(3

)1(2

)1(1

)(32

NPV Calculation

R=10%

A B

Year CF CF

0 -350 -250

1 50 125

2 100 100

3 150 75

4 200 50

NPV Calculation

• For A:

NPV(A)=27.4

• For B:

NPV(B)=36.78

OIR

CFnR

CFR

CFR

CFassetNPV

n/

)1(...

)1(3

)1(2

)1(1

)(32

350)1.1(

200

)1.1(

150

)1.1(

100

)1.1(

50)asset(NPV

432

250)1.1(

50

)1.1(

75

)1.1(

100

)1.1(

125)asset(NPV

432

The Bad: Internal rate of return• IRR is the minimum return (yield) on a real

investment so that the present value of the future cash flows is equal to the I/O--It is the (break-even) rate that sets NPV equal to zero.

• IRR=Additional cents on the $ invested• It assumes that CFs are reinvested at IRR• It might include several (irrelevant) solutions• It might provide contradictory results with NPV

nIRRCFn

IRRCF

IRRCF

IRRCF

OI)1(

...)1(

3)1(

2)1(

1/

32

IRR CalculationR=10%

A B A-B

Year CF CF CF

0 -350 -250 -100

1 50 125 -75

2 100 100 0

3 150 75 75

4 200 50 150

IRR 12.91% 17.80% 8.1%???

NPV vs. IRR

• NPV and IRR will generally give us the same decision

• Exceptions– Non-conventional cash flows – cash flow signs

change more than once– Mutually exclusive projects

• Initial investments are substantially different

• Timing of cash flows is substantially different

Another Example – Non-conventional Cash Flows

• Suppose an investment will cost $90,000 initially and will generate the following cash flows:– Year 1: 132,000

– Year 2: 100,000

– Year 3: -150,000

• The required return is 15%.• Should we accept or reject the project?

NPV Profile

($10,000.00)

($8,000.00)

($6,000.00)

($4,000.00)

($2,000.00)

$0.00

$2,000.00

$4,000.00

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

Discount Rate

NP

V

IRR = 10.11% and 42.66%

Summary of Decision Rules

• The NPV is positive at a required return of 15%, so you should Accept

• If you use the financial calculator, you would get an IRR of 10.11% which would tell you to Reject

• You need to recognize that there are non-conventional cash flows and look at the NPV profile

IRR and Mutually Exclusive Projects

• Mutually exclusive projects– If you choose one, you can’t choose the other

– Example: You can choose to attend graduate school at either Harvard or Stanford, but not both

• Intuitively you would use the following decision rules:– NPV – choose the project with the higher NPV

– IRR – choose the project with the higher IRR

Example With Mutually Exclusive Projects

Period Project A

Project B

0 -500 -400

1 325 325

2 325 200

IRR 19.43% 22.17%

NPV 64.05 60.74

The required return for both projects is 10%.

Which project should you accept and why?

NPV Profiles

($40.00)

($20.00)

$0.00

$20.00

$40.00

$60.00

$80.00

$100.00

$120.00

$140.00

$160.00

0 0.05 0.1 0.15 0.2 0.25 0.3

Discount Rate

NP

V AB

IRR for A = 19.43%

IRR for B = 22.17%

Crossover Point = 11.8%

Conflicts Between NPV and IRR

• NPV directly measures the increase in value to the firm

• Whenever there is a conflict between NPV and another decision rule, you should always use NPV

• IRR is unreliable in the following situations– Non-conventional cash flows

– Mutually exclusive projects

Summary – Discounted Cash Flow Criteria• Net present value

– Difference between market value and cost– Take the project if the NPV is positive– Has no serious problems– Preferred decision criterion

• Internal rate of return– Discount rate that makes NPV = 0– Take the project if the IRR is greater than the required return– Same decision as NPV with conventional cash flows– IRR is unreliable with non-conventional cash flows or mutually

exclusive projects• Payback period

– Length of time until initial investment is recovered– Take the project if it pays back within some specified period– Doesn’t account for time value of money and there is an arbitrary

cutoff period

A Better Method: MIRR

• Assume that Cash Flows are reinvested at the opportunity cost rate.

• Bring all positive cash flows to the future=FV(Positive cash flows)

• Bring all negative cash flows to the present =PV(Negative cash flows)

• Then, • FV(Positive cash flows)= PV(Negative cash flows) x FVIF(n, MIRR)

Example: MIRR

• For Project A

• Do Project B…

12%MIRR ,

58.1350

55.552FVIF

350CF-)(

55.552)1.1(200)1.1(150)1.1(100)1.1(50)(

MIRR)(4;

0123

Then

PV

CFFV

R=10%

A B

Year CF CF

0 -350 -250

1 50 125

2 100 100

3 150 75

4 200 50

The Ugly: Payback• Payback: length of time until the sum of an investment’s cash

flows equals its cost.Year CF Cumulated CF1 200 2002 400 6003 600 1200

I/O=$1,000Payback=2 year + 400/600=2 2/3 year• No time value• No risk• Focuses on liquidity; thus, biased against long term projects

• What is the most common measure of profitability in corporate America?

Payback CalculationR=10%

A B

Year CF CF

0 -350 -250

1 50 125

2 100 100

3 150 75

4 200 50

Payback 3.25 years 2.33 years

Capital Budgeting

• Capital budgeting– Cash flow

• Start form nothing=CFA• Expand or Replace=ΔCFA

– Cost of capital

Cash Flows That Matters...

• Stand-alone principle:

• Cash flow that matters in a new project: Cash flow from assets

• Cash flow that matters in a replacement or expansion project: Incremental Cash flow from assets

• Also,

spending) (-

spending) (NWC-

flow)cash (operatingassets) from flow (

Capital

Cash

Incremental Cash Flow Analysis (case of replacement or expansion Project) Δ revenues+ Δ costs (“-” for an increase in costs, “+” for savings in costs)

+ Δ Depreciation (“+” for an increase in DPR, “-” for a decrease in DPR)

+ Δ taxes (“-” for an increase in taxes, “+” for savings in taxes)

+ Δ NWC sp.(“-” for an increase in NWC sp., “+” for a decrease in NWC sp.)

+ Δ Fixed Assets spending (“-” for an increase in FA sp., “+” for a decrease in FA sp.)

---------------------------------------

Incremental (Δ )Cash flow from assets

Costs that matter…or not• Sunk costs (R&D, consulting fee)• Opportunity cost and externalities: cost of using a

rented vs. own building space (opportunity cost: you could lease/rent it for a certain amount of dollar)

• NWC: it is recovered at the end (2 techniques)• Terminal value (the value at the end…)• Initial outlay• Financing costs

– Are they included in “cash flow from assets”?– Would you consider them in evaluating the profitability

of a project? Why? How?

More Complicated Case:REPLACEMENT PROJECT

Ex: you are looking at replacing an old processing system with a new one. Installation costs of the new system (net of taxes) are $485,000, which is going to be depreciated to zero over five years. The new system can be scrapped for $60,000. The pre-tax operating cost savings are $100,000 per year. also, the new system requires an initial net working capital injection of $50,000. The tax rate is approximately 34%. The discount rate for this project is 15%. Go or no-go with the replacement?

An other replacement Problem

• Original Machine– Initial cost = 100,000

– Annual depreciation = 9000

– Purchased 5 years ago

– Book Value = 55,000

– Salvage today = 65,000

– Salvage in 5 years = 10,000

• New Machine– Initial cost = 150,000– 5-year life– Salvage in 5 years = 0– Cost savings = 50,000

per year– 3-year MACRS

depreciation

• Required return = 10%• Tax rate = 40%

Cost Cutting…• Your company is considering a new computer system that will

initially cost $1 million. It will save $300,000 a year in inventory and receivables management costs. The system is expected to last for five years and will be depreciated using 3-year MACRS. The system is expected to have a salvage value of $50,000 at the end of year 5. There is no impact on net working capital. The marginal tax rate is 40%. The required return is 8%.

More Complicated Case:EXPANSION PROJECT

Ex: HEP a tech company is looking at a full scale production of its “atomic filtration” device (AFD). The marketing department estimates that an additional 15,000 units can be sold at $2,000 a piece. Additional Equipment needed would cost $9.5 million and $0.5 million in installation. This equipment can be depreciated straight line in 5 years to zero. Initial net working capital injection is $4 million. The life of the project is 4 years at the end of which it can be sold for $2 million. Variable cost are not changing from prior the expansion—i.e., 60% of sales. However, fixed costs will increase at least by $5 million a year. The marginal tax rate is approximately 40%. The discount rate for this project is 15%. Go or no-go with the expansion?

Sensitivity Analysis/ Simulation

• A probability function for NPV

• Simulation Value at risk?

• Sensitivity analysis What variables really matter?

• Crystal ball example

• Real Options