The Time Value of Money
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 amount and future
amount 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
Time 0 1 2 3 4 5
5%
Cash Flows
-100 105
Outflow Inflow
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 x 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
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
-100
Interest 5.00 5.25 5.51 5.79 6.08
Total Value 105.00 110.25 115.76 121.55 127.63
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) 4% 5% 6%
1 1.0400 1.0500 1.0600
2 1.0816 1.1025 1.1236
3 1.1249 1.1576 1.1910
4 1.1699 1.2155 1.2625
5 1.2167 1.2763 1.3382
6 1.2653 1.3401 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 1 2 3 4 5
5%
Cash
-100 5.00 5.25 5.51 5.79 6.08
Flows
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
5 Future Value of $1
4 i= 15%
3
i= 10%
2 i= 5%
1
i= 0%
0
1 2 4 6 8 10
0 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
FVn 1
PV FVn n
1 i
n
(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 4 5
PV = ? ÷ 1.05 ÷ 1.05 ÷ 1.05 ÷ 1.05 127.63
-100.00 105.00 110.25 115.76 121.55
$127.63 $127.63
PV $127.63(0.7835) $100
1.05 5
1.2763
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
i= 15%
0 2 4 6 8 10 12 14 16 18 20
Periods
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 3
100 100 100.00 = 100 (1.05)0
105.00 = 100 (1.05)1
110.25 = 100 (1.05)2
315.25
Future Value of an Annuity
n -1
FVA n PMT(1 i)0 PMT(1 i)1 PMT(1 i)n -1 PMT 1 i t
t 0
n nt 1 i n 1
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 Ordinary
Annuity
$100 at the end of each year
0 5% 1 2 3
100 100 100.00 = 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 5% 1 2 3
100 100 100
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 5% 1 2 3
100 100 100
100
95.238
1.05 1
100
90.703
1.05 2
100
86.384
1.05 3
272.325
Present Value of an Annuity
Numerical solution:
1 1 1
PVA n PMT 1
PMT 2
PMT 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 1
PMT 2
PMT n
1 i
1 i
1 i
1
1
n 1 1 i n
PMT PMT
t 1 1 i
t i
1
1
1.05 3
$100 $100(2.7232) $272.32
0.05
Present Value of an Annuity
Financial calculator solution:
Inputs: N = 3 I = 5 PMT = -100 FV
= 0 PV = ?
Output: = 272.325
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 5% 1 2 3
100 100
1.05 100.000 100 100
1.05 1
1.05 0
100 100
1.05 95.238
1.05 1
1.05 1
100 100
1.05 90.703
1.05 2
1.05 2
285.941
Present Value of
an Annuity Due
Numerical solution:
n -1 1 n 1
PVA(DUE) n PMT PMT 1 i
t 0 1 i t 1 1 i
1 t
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
Perpetuities
Perpetuity - a stream of equal payments
expected to continue forever
Consol - a perpetual bond issued by the
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 1
CF2 2
CFn n
1 i 1 i 1 i
n 1
CFt t
t 1
1 i
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
CFt 1 i n - t
t 1
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
Annual Percentage Rate (APR)
the periodic rate multiplied by the number of
periods per year
this is not adjusted for compounding
More frequent compounding:
mn
i simple
FVn PV 1
m
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% 1 2 3
15,000 PMT PMT PMT
PMT PMT PMT
PVA 3
1 i 1
1 i 2
1 i 3
3 PMT
t 1 1 i t
3 PMT
$15,000
t 1 1.08 t
Amortized Loans
Numerical Solution:
1
1-
1 1.08
3 3 3
PMT
$15,000 PMT t
PMT
t 1 1.08 t 1 1.08
t
0.08
$15,000 PMT 2.5771
$15,000
PMT $5,820.50
2.5771
Amortized Loans
Financial calculator solution:
Inputs: N = 3 I = 8 PV = 15000 FV
= 0 PMT = ?
Output: = -5820.5
Amortized Loans
Repayment of Remaining
Amortization Schedule shows how a loan will
Year
Beginning
Amount (1)
Payment (2) Interest a (3) Principalb (2)- Balance (1)-
(3)=(4) (4)=(5)
1 be$ repaid with
15,000.00 $ a5,820.50
breakdown
$ of interest
1,200.00 $ and$
4,620.50 10,379.50
2 10,379.50 5,820.50 830.36 4,990.14 5,389.36
3 principle on each
5,389.36 payment431.15
5,820.50 date 5,389.35 0.01 c
a
Interest 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).
b
Repayment of principal is equal to the payment of $5,820.50 minus
the interest charge for each year.
c
The $0.01 remaining balance at the end of Year 3 results from
rounding differences.
