The Time Value of Money
Chapter 9
The Time Value of Money
 Which would you rather have ?
 $100 today - or
 $100 one year from today
 Sooner is better !
The Time Value of Money
 How about $100 today or $105 one year
from today?
 We revalue current dollars and future
dollars using the time value of money
 Cash flow time line graphically shows the
timing of cash flows
Cash Flow Time Lines
 Time 0 is today; Time 1 is one period from today
Interest rate
0
1
5%
Time
Cash Flows
-100
Outflow
105
Inflow
2
3
4
5
Future Value
 Compounding
 the process of determining the value of a
cash flow or series of cash flows some time
in the future when compound interest is
applied
Future Value
 PV = present value or starting amount,
say, $100
 i = interest rate, say, 5% per year would
be shown as 0.05
 INT = dollars of interest you earn during
the year $100  0.05 = $5
 FVn = future value after n periods or
$100 + $5 = $105 after one year
 = $100 (1 + 0.05) = $100(1.05) = $105
Future Value
FV1  PV  INT
 PV  PV(i)
 PV(1  i)
Future Value
 The amount to which a cash flow or
series of cash flows will grow over a given
period of time when compounded at a
given interest rate
Compounded Interest
 Interest earned on interest
FVn  PV(1  i)
n
Cash Flow Time Lines
Time
0 5%
1
2
3
4
5
5.00
5.25
5.51
5.79
6.08
Total Value 105.00
110.25
-100
Interest
115.76 121.55 127.63
Future Value Interest Factor
for i and n (FVIFi,n)
 The future value of $1 left on deposit for
n periods at a rate of i percent per period
 The multiple by which an initial
investment grows because of the interest
earned
Future Value Interest Factor
for i and n (FVIFi,n)
 FVn = PV(1 + i)n = PV(FVIFi,n)
Period (n)
1
2
3
4
5
6
4%
1.0400
1.0816
1.1249
1.1699
1.2167
1.2653
5%
6%
1.0500
1.1025
1.1576
1.2155
1.2763
1.3401
1.0600
1.1236
1.1910
1.2625
1.3382
1.4185
For $100 at i = 5% and n = 5 periods
Future Value Interest Factor
for i and n (FVIFi,n)
 FVn = PV(1 + i)n = PV(FVIFi,n)
Period (n)
1
2
3
4
5
6
4%
1.0400
1.0816
1.1249
1.1699
1.2167
1.2653
5%
6%
1.0500
1.1025
1.1576
1.2155
1.2763
1.3401
1.0600
1.1236
1.1910
1.2625
1.3382
1.4185
For $100 at i = 5% and n = 5 periods
$100 (1.2763) = $127.63
Financial Calculator
Solution
 Five keys for variable input
 N = the number of periods
 I = interest rate per period
may be I, INT, or I/Y
 PV = present value
 PMT = annuity payment
 FV = future value
Two Solutions
 Find the future value of $100 at 5%
interest per year for five years
 1. Numerical Solution:
Time
0
5%
Cash
-100
Flows
1
2
3
5.00
5.25
5.51
4
5.79
5
6.08
Total Value 105.00 110.25 115.76 121.55 127.63
FV5 = $100(1.05)5 = $100(1.2763) = $127.63
Two Solutions
2. Financial Calculator Solution:
Inputs: N = 5 I = 5 PV = -100 PMT = 0 FV = ?
Output: = 127.63
Graphic View of the Compounding
Process: Growth
 Relationship among Future Value,
Growth or Interest Rates, and Time
Future Value of $1
5
i= 15%
4
i= 10%
3
i= 5%
2
1
0
1
0
2
4
6
i= 0%
8
10
Periods
Present Value
 Opportunity cost
 the rate of return on the best available
alternative investment of equal risk
 If you can have $100 today or $127.63 at
the end of five years, your choice will
depend on your opportunity cost
Present Value
 The present value is the value today of a
future cash flow or series of cash flows
 The process of finding the present value
is discounting, and is the reverse of
compounding
 Opportunity cost becomes a factor in
discounting
Cash Flow Time Lines
0
1
2
3
4
5
5%
PV = ?
127.63
Present Value
 Start with future value:
 FVn = PV(1 + i)n
 1 
FVn
PV 
 FVn 
n
n
(1  i)
 1  i  
Two Solutions
 Find the present value of $127.63 in five years
when the opportunity cost rate is 5%
 1. Numerical Solution:
0
5% 1
2
3
PV = ? ÷ 1.05
÷ 1.05
÷ 1.05
-100.00 105.00
110.25
115.76
4
÷ 1.05
5
127.63
121.55
$127.63 $127.63
PV 

 $127.63(0.7835)  $100
5
1.2763
1.05
Two Solutions
 Find the present value of $127.63 in five
years when the opportunity cost rate is
5%
 2. Financial Calculator Solution:
Inputs: N = 5 I = 5 PMT = 0 FV = 127.63 PV = ?
Output: = -100
Graphic View of the
Discounting Process
Present Value of $1
 Relationship among Present Value,
1 Interest Rates, and Time
i= 0%
0.8
0.6
i= 5%
0.4
i= 10%
0.2
0
2
4
6
8
i= 15%
10
12
14
16
18
Periods
20
Solving for Time and
Interest Rates
 Compounding and discounting are
reciprocals
 FVn = PV(1 + i)n
 1 
FVn
PV 
 FVn 
n
n
(1  i)
 1  i  
Four variables: PV, FV, i and n
If you know any three, you can solve for the fourth
Solving for i
 For $78.35 you can buy a security that
will pay you $100 after five years
 We know PV, FV, and n, but we do not
know i
0 i=?
-78.35
1
2
3
4
FVn = PV(1 + i)n
$100 = $78.35(1 + i)5 Solve for i
5
100
Numerical Solution
 FVn = PV(1 + i)n
 $100 = $78.35(1 + i)5
$100
5
1  i  
 1.2763
$78.35
1
5
1  i   1.2763  1.05
i  1.05 - 1  0.05
Financial Calculator
Solution
 Inputs: N = 5 PV = -78.35 PMT = 0
FV = 100 I = ?
 Output: = 5
 This procedure can be used for any rate
or value of n, including fractions
Solving for n
 Suppose you know that the security will
provide a return of 10 percent per year,
that it will cost $68.30, and that you will
receive $100 at maturity, but you do not
know when the security matures. You
know PV, FV, and i, but you do not know
n - the number of periods.
Solving for n
 FVn = PV(1 + i)n
 $100 = $68.30(1.10)n
 By trial and error you could substitute
for n and find that n = 4
0 10%
-68.30
1
2
n-1
n=?
100
Financial Calculator
Solution
 Inputs: I = 10 PV = -68.30 PMT = 0
FV = 100 N = ?
 Output: = 4.0
Annuity
 An annuity is a series of payments of an equal
amount at fixed intervals for a specified
number of periods
 Ordinary (deferred) annuity has payments at
the end of each period
 Annuity due has payments at the beginning of
each period
 FVAn is the future value of an annuity over n
periods
Future Value of an Annuity
 The future value of an annuity is the
amount received over time plus the
interest earned on the payments from the
time received until the future date being
valued
 The future value of each payment can be
calculated separately and then the total
summed
Future Value of an Annuity
 If you deposit $100 at the end of each year for
three years in a savings account that pays 5%
interest per year, how much will you have at the
end of three years?
0
5%
1
2
100
100
3
100.00 = 100 (1.05)0
105.00 = 100 (1.05)1
110.25 = 100 (1.05)2
315.25
Future Value of an Annuity
FVA n  PMT(1  i)0  PMT(1  i)1    PMT(1  i)n -1  PMT
n -1

t 0
1  i t
 1  i n  1
n
nt 

 PMT  1  i 
 PMT 

t 1

i


 1.053  1
FVA 3  $100
  $100(3.1525)  $315.25
 0.05 
Future Value of an Annuity
 Financial calculator solution:
 Inputs: N = 3 I = 5 PV = 0 PMT = 100 FV = ?
 Output: = 315.25
 To solve the same problem, but for the
present value instead of the future value,
change the final input from FV to PV
Annuities Due
 If the three $100 payments had been
made at the beginning of each year, the
annuity would have been an annuity due.
 Each payment would shift to the left one
year and each payment would earn
interest for an additional year (period).
Future Value of an Annuity
 $100 at the end of each year
0
5%
1
2
100
100
3
100.00 = 100 (1.05)0
105.00 = 100 (1.05)1
110.25 = 100 (1.05)2
315.25
Future Value of
an Annuity Due
 $100 at the start of each year
0
100
5%
1
2
100
100
3
105.00
= 100 (1.05)1
110.25
= 100 (1.05)2
115.7625 = 100 (1.05)3
331.0125
Future Value of
an Annuity Due
 Numerical solution:
n
t
FVA(DUE) n  PMT  1  i  
 t 1

 n

n -t 
 PMT  1  i    1  i 

 t 1

 1  i n  1

 PMT 
  1  i 
i



Future Value of
an Annuity Due
 Numerical solution:
  1.053  1

FVA(DUE)n  $100 
  1.05
  0.05 

 $1003.1525  1.05
 $331.0125
Future Value of
an Annuity Due
 Financial calculator solution:
 Inputs: N = 3 I = 5 PV = 0 PMT = 100 FV = ?
 Output: = 331.0125
Present Value of an Annuity
 If you were offered a three-year annuity
with payments of $100 at the end of each
year
 Or a lump sum payment today that you
could put in a savings account paying 5%
interest per year
 How large must the lump sum payment
be to make it equivalent to the annuity?
Present Value of an Annuity
0
100
 95.238
1
1.05
100
 90.703
2
1.05
100
 86.384
3
1.05
272.325
5%
1
2
3
100
100
100
Present Value of an Annuity
 Numerical solution:
 1 
 1 
 1 
PVA n  PMT 
 PMT 
   PMT 
1
2
n
 1  i  
 1  i  
 1  i  
n
1 
 PMT  

t  1 1  i t


Present Value of an Annuity
 1 
 1 
 1 
PVA n  PMT 
 PMT 
   PMT 
1
2
n






1

i
1

i
1

i






1 

1

n
1 

1  i n 

 PMT  
  PMT 
t
i


 t 1 1  i  


1 

1



1.05 3 
  $100(2.7232)  $272.32
 $100 
 0.05 


Present Value of an Annuity
 Financial calculator solution:
 Inputs: N = 3 I = 5 PMT = -100
= 0 PV = ?
 Output: = 272.325
FV
Present Value of
an Annuity Due
 Payments at the beginning of each year
 Payments all come one year sooner
 Each payment would be discounted for
one less year
 Present value of annuity due will exceed
the value of the ordinary annuity by one
year’s interest on the present value of the
ordinary annuity
Present Value of
an Annuity Due
0
100
 1.05 
1
100
1.05
1.05
100
100
 1.05 
1
1.05
100
0
1.05
1
5%
1
 100.000 100
 95.238
100


 1.05 
 90.703
2
2
1.05
1.05
285.941
2
100
3
Present Value of
an Annuity Due
 Numerical solution:
 n

 n -1 1 
1 
PVA(DUE)n  PMT  
  1  i 
  PMT   
1
t
 t  0 1  i  
 t 1 1  i  

1 


1



n


1

i


 PMT  
  1  i 
i










Present Value of
an Annuity Due
1 


 1 

3

1.05 


PV(DUE)3  $100 
  1.05
  0.05 







 $100 [(2.72325)(1.05)]
 $100 (2.85941)
 $285.941
Present Value of
an Annuity Due
 Financial calculator solution:
 Switch to the beginning-of-period mode,
then enter
 Inputs: N = 3 I = 5 PMT = -100 FV
= 0 PV = ?
 Output: = 285.94
 Then switch back to the END mode
Solving for Interest Rates
with Annuities
 Suppose you pay $846.80 for an
investment that promises to pay you $250
per year for the next four years, with
0 payments
1 made at 2the end of3each year4
i=?
-846.80
1 

1

250
250

1 250
i 4 

$846.80  $250 
i




250
Solving for Interest Rates
with Annuities
 Numerical solution:
 Trial and error using different values for
i using until you find i where the present
value of the four-year, $250 annuity
equals $846.80. The solution is 7%.
Solving for Interest Rates
with Annuities
 Financial calculator solution:
 Inputs: N = 4 PV = -846.8 PMT = 250
FV = 0 I = ?
 Output: = 7.0
Perpetuities
 Perpetuity - a stream of equal payments
expected to continue forever
 Consol - a perpetual bond issued by the
British government to consolidate past
debts; in general, and perpetual bond
Payment
PMT
PVP 

Interest Rate
i
Uneven Cash Flow Streams
 Uneven cash flow stream is a series of
cash flows in which the amount varies
from one period to the next
 Payment (PMT) designates constant cash
flows
 Cash Flow (CF) designates cash flows in
general, including uneven cash flows
Present Value of
Uneven Cash Flow Streams
 PV of uneven cash flow stream is the sum
of the PVs of the individual cash flows of
the stream
 1 
 1 
 1 
PV  CF1 
 CF2 
   CFn 
1
2
n
 1  i  
 1  i  
 1  i  
 1 
  CFt 
t
t 1
 1  i  
n
Future Value of
Uneven Cash Flow Streams
 Terminal value is the future value of an
uneven cash flow stream
FVn  CF1 1  i n -1  CF2 1  i n - 2    CFn 1  i 0

n

t 1
CFt 1  i n - t
Solving for i with
Uneven Cash Flow Streams
 Using a financial calculator, input the CF
values into the cash flow register and
then press the IRR key for the Internal
Rate of Return, which is the return on
the investment.
Compounding Periods
 Annual compounding
 interest is added once a year
 Semiannual compounding
 interest is added twice a year
 10% annual interest compounded semiannually

would pay 5% every six months
adjust the periodic rate and number of periods
before calculating
Interest Rates
 Simple (Quoted) Interest Rate
 rate used to compute the interest payment paid
per period
 Effective Annual Rate (EAR)
 annual rate of interest actually being earned,
considering the compounding of interest
m
i simple 

  1.0
EAR   1 
m 

Interest Rates
 Annual Percentage Rate (APR)
 the periodic rate multiplied by the number of
periods per year
this is not adjusted for compounding

 More frequent compounding:
i simple 


FVn  PV  1 
m 

mn
Amortized Loans
 Loans that are repaid in equal payments
over its life
 Borrow $15,000 to repay in three equal
payments at the end of the next three
years, with 8% interest due on the
outstanding loan balance at the
beginning of each year
Amortized Loans
0
8%
15,000
1
2
PMT
PMT
PVA 3 

$15,000 
PMT
1  i 
1
PMT

1  i 
2
3
PMT
t 1
1  i t
3
PMT
t 1
1.08


t
3
PMT

PMT
1  i 3
Amortized Loans
 Numerical Solution:
1 

13
3
3


PMT
1 

1.08 

$15,000  
 PMT 
 PMT 
t
t 
 0.08 
t 1 1.08
 t 1 1.08 


$15,000  PMT 2.5771
$15,000
PMT 
 $5,820.50
2.5771
Amortized Loans
 Financial calculator solution:
 Inputs: N = 3 I = 8 PV = 15000 FV =
0 PMT = ?
 Output: = -5820.5
Amortized Loans
Repayment of Remaining
Beginning
a
Payment (2) Interest (3) Principalb (2)- Balance (1)Amount (1)
(4)=(5)
(3)=(4)
 Amortization Schedule shows how a loan will
1
$ repaid
15,000.00 with
$
5,820.50
$
1,200.00of $interest
4,620.50
$
10,379.50
be
a
breakdown
and
2
10,379.50
5,820.50
830.36
4,990.14
5,389.36
3
5,389.36
5,820.50
431.15
5,389.35
0.01
principle on each payment date
Year
aInterest
is calculated by multiplying the loan balance at the beginning
of the year by the interest rate. Therefor, interest in Year 1 is
$15,000(0.08) = $1,200; in Year 2, it is $10,379.50(0.08)=$830.36;
and in Year 3, it is $5,389.36(0.08) = $431.15 (rounded).
bRepayment
of principal is equal to the payment of $5,820.50 minus
the interest charge for each year.
cThe
$0.01 remaining balance at the end of Year 3 results from
rounding differences.
c
Comparing Interest Rates
 1. Simple, or quoted, rate, (isimple)
 rates compare only if instruments have the
same number of compounding periods per
year
 2. Periodic rate (iPER)
 APR represents the periodic rate on an
annual basis without considering interest
compounding
 APR is never used in actual calculations
Comparing Interest Rates
 3. Effective annual rate, EAR
 the rate that with annual compounding (m=1)
would obtain the same results as if we had used the
periodic rate with m compounding periods per
year
m
i SIMPLE 

EAR   1 
  1.0
m 

m
 1  i PER   1
End of Chapter 9
The Time
Value of
Money