Present and Future Value
Translating cash flows
forward and
backward through
time
Future Value
FV  P(1  r )
• Money invested earns interest
and interest reinvested earns
more interest
• The power of compounding
T
Future Value Problems
FV  P(1  r )
T
Solve for any variable, given the other three
•
•
•
•
FV: How much will I have in the future?
P: How much do I need to invest now?
r: What rate of return do I need to earn?
T: How long will it take me to reach my goal?
Present Value
FV
P
T
(1  r )
• Discounting future cash flows at the
“opportunity cost” (cost of capital, discount
rate, minimum acceptable return)
• A dollar tomorrow is worth less than a dollar
today
Present Values can be Added
CF1
CF2
CFT
P  CF0 

 ... 
2
(1  r ) (1  r )
(1  r )T
 1 
 1
 1 
CF2  ...  
 CF0  
CF1  
2 
T
1

r
(
1

r
)
(
1

r
)






CFT

• Cash flows further out are discounted more
• Discount factors are like prices (exchange
rates)
Calculating PV of a Stream (Beware)
• Calculator assumes
first CF you give it
occurs now (Time 0)
• Excel assumes first
CF you give it
occurs one year
from now (Time 1)
Different Compounding Periods
APR 

(1  EAR)  1 

m 

m
• m = # of compounding periods in a year
• APR = actual rate x m (APR is annualized)
• EAR = the annually compounded rate that
gives the same proceeds as APR
compounded m times
Semiannual Compounding
2
.10 

1 
  1.1025
2 

• m=2
• APR = 10%
• EAR = 10.25%
Quarterly Compounding
4
.10 

1 
  1.1038
4 

• m=4
• APR = 10%
• EAR = 10.38%
Monthly Compounding
12
 .10 
1 
  1.1047
 12 
• m = 12
• APR = 10%
• EAR = 10.47%
Daily Compounding
.10 

1 

365 

• m = 365
• APR = 10%
• EAR = 10.516%
365
 1.10516
Continuous Compounding
m
 APR 
APR
1 
  e as m  
m 

e
.10
 1.10517
• m=
• APR = 10%
• EAR = 10.517%
Annuities
• All cash flows are the same, so we can factor
out the constant payment C and calculate the
sum of the discount factors
C
C
C
P

 ... 
2
T
1  r (1  r )
(1  r )
 1
1
1
 C 

 ... 
2
T
(1  r )
 1  r (1  r )



Special Case: Perpetuity
• If all the cash flows are the same each period
forever, the sum of the discount factors
converges to 1/r
 1

1
1
P  C 


 ...
2
3
 1  r (1  r ) (1  r )

C

r
Perpetuity Example
• Let C = $100 and r = .05
• $100 per year forever at 5% is worth:
100
P
 2000
.05
Other Perpetuity Examples
• British Consol
Bonds
• Canadian Pacific 4%
Perpetual Bonds
• Endowments
– How much can I
withdraw annually
without invading
principal?
Growing Perpetuity
• Suppose the initial payment C grows at a
constant rate g per period (where g < r)
• This growing stream still has a finite present
value:
C
C (1  g ) C (1  g )
P


 ...
2
3
(1  r ) (1  r )
(1  r )
C

rg
2
Growing Perpetuity Example
• Suppose the initial payment is $100 and that
this grows at 3% per year while the discount
rate is 5%
• The value of this growing perpetuity is:
C
100
P

 $5,000
r  g .05  .03
Other Growing Perpetuity Examples
• Stock price = present
value of growing
dividend stream (see
Class #7)
• M&A: How to estimate
terminal value
– How fast do earnings
grow after the end of
the analysis period?
Finite Annuity=
Difference Between Two Perpetuities
0
C
1
C
2
C
3
C
4
C
5
C
6
C
7
C
8
C
C
C
C
C
P
r
P
C 1 


4 
r  (1  r ) 
C
1 
difference  1 
r  (1  r )4 
Annuity Example
• What’s the value of a 4-year annuity with
annual payments of $40,000 per year (@5%)?
C
P
r

1 
1  (1  r ) 4  


40,000 
1 
1
 141,838

4
.05  (1.05) 
Oops, Tuition Payments Due at
Beginning of Year
 1

1

P  C 1  1 
T 1  
 r  (1  r ) 

1 
1 
1 

 40,000 1 
3 
 .05  (1.05) 
 148,930  141,838(1.05)
Other Annuity Applications
• Lottery winnings
• Lease & loan
contracts
• Home mortgages
• Retirement savings/
income
Home Mortgages
• 30-year fixed rate mortgage: 360 equal
monthly payments
• Most of early payments goes toward
interest; principal repayment gradually
accelerates
• At any point: outstanding balance =
present value of remaining payments
More Annuity Problems
Saving, Retirement Planning,
Evaluating Loans and
Investments
Net Present Value (NPV)
• Best criterion for
corporate
investment:
• Invest if NPV > 0
NPV with a Single,
Initial Investment Outlay
T
Ct
NPV  

I
t
t 1 (1  r )
•
•
•
•
I = initial investment outlay
Ct = project cash flow in period t
r = discount rate (shareholders’ opp. cost)
T = project termination period
Implications of NPV > 0
T
Ct

I

t
t 1 (1  r )
•
•
•
•
Project benefits exceed cost (in PV terms)
Project is worth more than it costs
Project market value exceeds book value
Project adds shareholder value
NPV More Generally
T
Ct
NPV  
t
t  0 (1  r )
• Treat inflows as +, outflows as –
• NPV = PV of all cash flows
• Investment may occur throughout project life
Internal Rate of Return
T
Ct

I

t
t 1 (1  IRR )
• IRR sets value of benefits = investment
• IRR sets NPV = 0
• IRR is the rate of return company expects on
investment I
NPV > 0 Implies IRR > r
T
Ct
NPV  0  

I
t
t 1 (1  r )
• If NPV > 0, IRR must exceed r
• Investing when NPV > 0 implies company
expects to earn more than shareholder’ opp.
Cost
• Equivalent: Invest when NPV > 0 or when
IRR>I