THE TIME VALUE OF MONEY
Chapter 3
WHAT COMPANIES DO
Transports of delight?
Meyrick and Associates, a consulting group, together with EconSearch
and Steer Davies Gleave, presented a report to the Victorian Government on
a range of transport options for the East–West Link.
The report summarised extensive analyses undertaken to evaluate the
benefits and costs of developing the new transport infrastructure options,
and described a series of present values that had been determined.
The present values were determined by:
1. estimating revenue per day for the base case and for each option
2. estimating revenue per year for each option
3. discounting the future cash flows
4. estimating the yearly increase of revenue for each option
5. calculating the difference between the base case and each option.
THE TIME VALUE OF MONEY
Financial managers compare the marginal benefits
and marginal costs of investment projects.
Projects usually have a long-term horizon: the
timing of benefits and costs matters.
The time value of money: a dollar received today is
worth more than a dollar received in the future.
FUTURE VALUE
Future value: the value of an investment made today
measured at a specific future date using compound interest.
FVn = PV×(1+r)n
Future value
depends on:
Interest rate
Number of periods
Compounding interval
PRESENT VALUE
Present value: the value today of a cash flow to be received
at a specific date in the future, assuming an opportunity to
earn interest at a specified rate.
FVn  PV  1  r 
n
PV 
FVn
(1  r )
n
FUTURE VALUE OF CASH
FLOW STREAMS
Mixed
stream
• A series of unequal cash flows
reflecting no particular pattern.
Annuity
• A stream of equal periodic cash
flows.
n
FV   CFt  1 r
t1
nt
(1  r ) n  1
FV  PMT 
 $5750.74
r
(1  r ) n  1
FV  PMT 
 1  r   $6153.29
r
PRESENT VALUE OF CASH
FLOW STREAMS
• Mixed streams
• Annuities
• Perpetuities: cash flow streams that continue
forever
n
PV   CFt 
t1
1
1 r
t
PRESENT VALUE OF A
PERPETUITY
• For a constant stream of cash flows that
continues forever

1
PV  PMT  
t
(
1

r
)
t 1
1
 PMT 
r
PMT

r
PRESENT VALUE OF A
GROWING PERPETUITY
CF1
PV0 
rg
0
1

$1000
$1000(1+0.02)1
$1000
$1020
2
$1000(1+0.02)2
$1040.4
rg
3
4
$1000(1+0.02)3 …
$1061.2
Growing perpetuity
CF1 = $1000
r = 7% per year
g = 2% per year
$1000
PV0 
 $20 000
0.07  0.02
COMPOUNDING MORE
FREQUENTLY THAN ANNUALLY
• m compounding periods
r

FVn  PV  1  
 m
• Continuous compounding
mn
 
FVn  PV  e
r n
• The more frequent the compound period, the
larger the FV!
COMPOUNDING MORE
FREQUENTLY THAN ANNUALLY
FV at end of two years of $125 000 at 5% interest
• Semiannual compounding:
 0.05 
FV2  $125 000  1 

2 

22
 $137 976.61
• Quarterly compounding:
 0.05 
FV2  $125 000  1 

4 

42
 $138 060.76
• Continuous compounding:
FV2  $125 000  e0.052  $138 146.365
STATED VERSUS EFFECTIVE
ANNUAL INTEREST RATES
Stated
annual rate
• The contractual annual rate of
interest charged by a lender or
promised by a borrower.
Effective
annual rate
• The annual rate of interest actually
paid or earned, reflecting the impact
of compounding frequency.
r 

EAR  1 

m

m
1
STATED VERSUS EFFECTIVE
ANNUAL INTEREST RATES
Average
annual
percentage
rate (AAPR)
Annual
percentage
yield (APY)
• The stated annual rate calculated by
multiplying the periodic rate by the
number of periods in one year.
• The annual rate of interest actually
paid or earned, reflecting the impact
of compounding frequency. The
same as the effective annual rate.