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
i nm
FV  PV  (1  )
m
• with factor
FV  PV  FVIF
i
nm;
m
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


FV



i  m 
 1

 PV 



or
FV
FVIF

i
( nm ; )
PV
m
1
nm
or
Input: FV - P V n Comput e i
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
1
1




nm
4
FV
3




i  m  
 1  2     1  64%

 PV 

 1 





FVIF
i
( 4; )
2
i
 3   32%  i  64%
2
3 FV - 1 P V 24 n CP T i  i  31.9%
T heni  31.9% 2  64%
Implicit time
FV
ln(
)
1
PV
n 
m ln(1  i )
m
or
FV
FVIF
i 
( nm ; )
PV
m
or
Input: FV - P V i Comput e n
Example
• You have $430 today. You need $500. If you
earn 1 percent per month, how many
months will you wait?
Solution
FV
500
ln(
)
ln(
)
1
1
PV
430  1.26  15m onths
n 
 
m ln(1  i ) 12 ln(1  12% )
m
12
or
500
FVIF(12 n;1%) 
 1.163  12n  15 m onths
430
or
500 FV - 430 P V 1 i Compute n
T henn  15 months
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:
1


1


i nm 
 (1  ) 
m


PVA  PMT 
i




m


• With factor:
PVA  PMT  PVIFA
nm;
i
m
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:
i nm 

(
1

)

1


m
FVA  PMT  

i


m


• with factor:
FVA  PMT  FVIFA
i
nm;
m
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
•
•
•
•
•
•
InsertFunctionFinancial
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)
CF 1
CF 2
CF 3
CFn
PV (asset ) 


 ... 
2
3
n
(1  R) (1  R) (1  R)
(1  R)
• 3 elements:
– CF=cash flow
– Maturity=n
– Interest rate=RAverage cost of moneyCost 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?
The good: Net Present Value
• Formula:
CF1
CF 2
CF 3
CFn
NPV (asset) 


 ... 
 I /O
2
3
n
(1  R) (1  R) (1  R)
(1  R)
• 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.
NPV Calculation
R=10%
Year
0
1
2
3
4
A
CF
-350
50
100
150
200
B
CF
-250
125
100
75
50
NPV Calculation
CF1
CF 2
CF 3
CFn
NPV (asset) 


 ... 
 I /O
2
3
n
(1  R) (1  R) (1  R)
(1  R)
• For A:
50
100
150
200
NP V(asset) 



 350
2
3
4
(1  .1) (1  .1)
(1  .1)
(1  .1)
NPV(A)=27.4
• For B:
NP V(asset) 
125
100
75
50



 250
2
3
4
(1  .1) (1  .1)
(1  .1)
(1  .1)
NPV(B)=36.78
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 (breakeven) rate that sets NPV equal to zero.
CF1
CF 2
CF 3
CFn
I /O 


 ... 
2
3
(1  IRR) (1  IRR) (1  IRR)
(1  IRR)n
• 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
IRR Calculation
R=10%
Year
0
1
2
3
4
IRR
A
CF
-350
50
100
150
200
12.91%
B
CF
-250
125
100
75
50
17.80%
A-B
CF
-100
-75
0
75
150
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 – Nonconventional 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
IRR = 10.11% and 42.66%
$4,000.00
$2,000.00
NPV
$0.00
($2,000.00)
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
($4,000.00)
($6,000.00)
($8,000.00)
($10,000.00)
Discount Rate
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 nonconventional 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
IRR for A = 19.43%
$160.00
IRR for B = 22.17%
$140.00
Crossover Point = 11.8%
$120.00
NPV
$100.00
$80.00
A
B
$60.00
$40.00
$20.00
$0.00
($20.00) 0
0.05
0.1
0.15
($40.00)
Discount Rate
0.2
0.25
0.3
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
FV (CF )  50  (1  .1) 3  100 (1  .1) 2  150 (1  .1)1  200 (1  .1) 0  552.55
PV (CF-)  350
552.55
 1.58
350
Then, MIRR  12%
FVIF(4;MIRR) 
• Do Project B…
R=10%
Year
0
1
2
3
4
A
CF
-350
50
100
150
200
B
CF
-250
125
100
75
50
The Ugly: Payback
• Payback: length of time until the sum of an investment’s cash
flows equals its cost.
YearCF
Cumulated CF
1
200
200
2
400
600
3
600
1200
I/O=$1,000
Payback=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 Calculation
R=10%
Year
0
1
2
3
4
Payback
A
CF
-350
50
100
150
200
3.25 years
B
CF
-250
125
100
75
50
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, (Cash flow from assets) (operatingcash flow)
- (NWCspending)
- (Capitalspending)
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 3year 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