Chapter 5
The Time Value of Money
Some Important Concepts
1
Topics Covered
Future Values and Compound Interest
 Present Values
 Multiple Cash Flows
 Perpetuities and Annuities
 Effective Annual Interest Rates.
 Inflation

2
Future Values
Future Value - Amount to which an investment
will grow after earning interest.
Compound Interest - Interest earned on interest.
Simple Interest - Interest earned only on the
original investment.
3
Future Values
Example - Simple Interest
Interest earned at a rate of 6% for five years
on a principal balance of $100.
Today
0
1
Interest Earned
Value
100
6
106
Future Years
2
3
4
5
6
112
6
118
6
124
6
130
Value at the end of Year 5 = $130
4
Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years
on the previous year’s balance.
Today
0
1
Interest Earned
Value
100
6
106
Future Years
2
3
4
5
6.36
6.74 7.15
7.57
112.36 119.10 126.25 133.82
Value at the end of Year 5 = $133.82
5
Future Values
Future Value of $100 = FV
FV  $100  (1  r )
t
6
Future Values
7000
0%
6000
5000
10%
4000
15%
3000
2000
1000
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
0
FV of $100
5%
Number of Years
7
Future Values
Example – Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1626.
Was this a good deal?
To answer, determine $24 is worth in the year 2006,
compounded at 8%.
FV  $24  (1  .08)
 $120.57 trillion
380
Financial calculator: n=380, i=8, PV=-24, PMT=0 
FV= $120.57 trillion
8
Set up the Texas instrument
2nd, “FORMAT”, set “DEC=9”, ENTER
 2nd, “FORMAT”, move “↓” several times,
make sure you see “AOS”, not “Chn”.
 2nd, “P/Y”, set to “P/Y=1”
 2nd, “BGN”, set to “END”



P/Y=periods per year,
END=cashflow happens end of periods
Present Values

If the interest rate is at 6 percent per year, how
much do we need to invest now in order to
produce $106 at the end of the year?
Present va lue (PV) 

future value $106

 $100
1.06
1.06
How much would we need to invest now to
produce $112.36 after two years?
Present va lue (PV) 
future value $112.36

 $100
1.06 2
1.06 2
10
Present Values
PV =
Future Value after t periods
PV: Discounted Cash-Flow
(1+ r) t
r: Discount Rate
(DCF)
DF 
1
( 1 r ) t

Discount Factor (DF) = PV of $1

Discount Factors can be used to compute the present
value of any cash flow.
11
Present Values
120
Interest Rates
PV of $100
5%
100
10%
80
15%
60
40
20
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
12
Present Values
Example
In 2007, Puerto Rico needed to borrow about $2.6 billion for up to
47 years. It did so by selling IOUs, each of which simply promised
to pay the holder $1,000 at the end of that time. The market
interest rate at the time was 5.15%. How much would you have
been prepared to pay for one of these IOUs?
1
PV  $1,000 
 $1,000  .0944  $94.4
47
(1.0515)
13
Present Values
Example
Kangaroo Autos is offering free credit on a $20,000
car. You pay $8,000 down and then the balance at
the end of 2 years. Turtle Motors next door does not
offer free credit but will give you $1,000 off the list
price. If the interest rate is 10%, which company is
offering the better deal?
14
Present Values
$12,000
$8,000
PV at Time=0
0
1
2
$8,000
$12,000 
Total
1
 $9,917.36
(1.10) 2
$17,917.36
15
Present Values
Example – Finding the Interest Rate
Each Puerto Rico IOU promised to pay holder $1,000 at
the end of 47 years. It is sold for $94.4. What is the
interest rate?
$94.4  (1  r) 47  $1,000
$1,000
(1  r) 
 10.593
$94.4
(1  r)  10.5931/47  1.0515
47
r  0.0515, or 5.15%
Financial calculator: n=47, FV=1,000 PV=-94.4, PMT=0
 i=5.15
16
Present Values
Example – Double your money
Suppose your investment advisor promises to double
your money in 8 years. What interest rate is implicitly
being promised?
FV  PV  (1  r) t
$2  $1 (1  r) 8
r  0.0905, or 9.05%
Financial calculator: n=8, FV=2 PV=-1, PMT=0  i=9.05
17
Present Value of Multiple Cash Flows
Example
Your auto dealer gives you the choice to pay $15,500 cash now,
or make three payments: $8,000 now and $4,000 at the end of
the following two years. If your cost of money is 8%, which do you
prefer?
Immediatepayment 8,000.00
PV1 
4 , 000
(1.08)1
 3,703.70
PV2 
4 , 000
(1.08) 2
 3,429.36
Total PV
 $15,133.06
18
What is the PV of this uneven
cash flow stream?
0
1
2
3
4
100
300
300
-50
10%
90.91
247.93
225.39
-34.15
530.08 = PV
Detailed steps (Texas Instrument
calculator)
To clear historical data:
 CF, 2nd ,CE/C
 To get PV:
 CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,
Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter,
↓,CPT
 “NPV=530.0867427”

Future Value of Multiple Cash Flows
Example
You plan to save some amount of money each year.
You put $1,200 in the bank now, another $1,400 at the
end of the first year, and a third deposit of $1,000 at
the end of the second year. If the interest rate is 8%
per year, how much will be available to spend at the
end of the 3 year?
21
Future Value of Multiple Cash Flows
$1,400
$1,200
$1,000
Year
0
1
2
3
FV in year 3
$1,080.00  $1,000 1.08
$1,632.96
 $1,400  (1.08) 2
$1,511.65
 $1,200  (1.08)3
$4,224.61
22
Perpetuities and Annuities

Annuity


Equally spaced level stream of cash flows
for a limited period of time.
Perpetuity

A stream of level cash payments that never
ends.
23
Perpetuities and Annuities
cash payment
C
PV of Perpertuit y  
r
interest rate
24
Perpetuities and Annuities
Example - Perpetuity
In order to create an endowment, which pays
$100,000 per year, forever, how much money must be
set aside today in the rate of interest is 10%?
100,000
PV 
 $1,000,000
0.10
25
Perpetuities and Annuities
Example - continued
If the first perpetuity payment will not be received until
three years from today, how much money needs to be
set aside today?
, 000
PV  1(1, 000
2  $826, 446
 0.10)
26
Perpetuities and Annuities
Cash Flow
Year
1. Perpetuity A
0
1
$1
2
$1
3
$1
2. Perpetuity B
3. Three-year annuity
$1
$1
$1
4
$1
5
$1
6 …
$1 …
$1
$1
$1 …
Present Value
1
r
1
r (1  r ) 3
1
1

r
r (1  r ) 3
27
Perpetuities and Annuities


1
1

PV of t - year annuity  C  
 r r (1  r )t 
C = cash payment
r = interest rate
t = Number of years cash payment is received
28
Perpetuities and Annuities
PV Annuity Factor (PVAF) - The present
value of $1 a year for each of t years.


1 
PVAF   1 
 r r (1  r )t 
29
Perpetuities and Annuities
Example – Lottery
Suppose you bought lottery tickets and won $295.7
million. This sum was scheduled to be paid in 25 equal
annual installments of $11.828 million each. Assuming
that the first payment occurred at the end of 1 year,
what was the present value of the prize? The interest
rate was 5.9 percent.
1
1
PV  $11.828  [

]
0.059 0.059(1.059) 25
PV  $152.6
Financial calculator: n=25, i=5.9, PMT=11.828, FV=0,  PV=-152.6
30
Perpetuities and Annuities

Annuity Due

Level stream of cash flows starting
immediately.
PV of an annuity due  (1  r)  PV of an annuity
31
Perpetuities and Annuities
Example – Lottery (Continued)
What is the value of the prize if you receive the first installment of
$11.828 million up front and the remaining payments over the
following 24 years?
1
1

] 1.059
25
0.059 0.059(1.059)
PV  $152.6 1.059  $161.7
PV  $11.828  [
32
Perpetuities and Annuities

Future Value of annual payments


t
1
1


FV  C  
 (1  r )
 r r (1  r )t 
(1  r ) t  1
C
r
33
Perpetuities and Annuities
Example - Future Value of annual payments
You plan to save $4,000 every year for 20 years and then retire.
Given a 10% rate of interest, what will be the FV of your retirement
account?


1
1
  (1  0.10) 20
FV  4,000  

 0.10 0.10(1  0.10) 20 
FV  $229,100
Financial calculator: n=20, i=10, PMT=-4,000, PV=0,
 FV=229,100
34
Effective Annual Interest Rate
Effective Annual Interest Rate - Interest rate that
is annualized using compound interest.
Annual Percentage Rate (APR) - Interest rate
that is annualized using simple interest.
35
Effective Annual Interest Rate
Example
Given a monthly rate of 1%, what is the Effective
Annual Rate(EAR)? What is the Annual Percentage
Rate (APR)?
EAR = (1 + 0.01)12 - 1 = 0.1268 or 12.68%
APR = 0.01 x 12 = 0.12 or 12.00%
36
Classifications of interest rates

1. Nominal rate (iNOM) – also called the APR,
quoted rate, or stated rate. An annual rate that
ignores compounding effects. Periods must
also be given, e.g. 8% Quarterly.

2. Periodic rate (iPER) – amount of interest
charged each period, e.g. monthly or quarterly.
 iPER
= iNOM / m, where m is the number of
compounding periods per year. e.g., m = 12 for
monthly compounding.
Classifications of interest rates

3. Effective (or equivalent) annual rate
(EAR, also called EFF, APY) : the annual
rate of interest actually being earned,
taking into account compounding.
 If the interest rate is compounded m
times in a year, the effective annual
interest rate is
m
 inom 
1 
 1
m 

Example, EAR for 10%
semiannual investment

EAR= ( 1 + 0.10 / 2 )2 – 1 = 10.25%
 An
investor would be indifferent
between an investment offering a
10.25% annual return, and one
offering a 10% return compounded
semiannually.
EAR on a Financial Calculator
Texas Instruments BAII Plus
keys:
description:
[2nd] [ICONV]
Opens interest rate conversion menu
Sets 2 payments per year
[↑] [C/Y=] 2 [ENTER]
Sets 10 APR.
[↓][NOM=] 10 [ENTER]
[↓] [EFF=] [CPT]
10.25
Why is it important to consider
effective rates of return?

An investment with monthly payments is
different from one with quarterly
payments.
 Must use EAR for comparisons.
 If iNOM=10%, then EAR for different
compounding frequency:
Annual
Quarterly
Monthly
Daily
10.00%
10.38%
10.47%
10.52%
What’s the FV of a 3-year $100
annuity, if the quoted interest rate is
10%, compounded semiannually?
0
1
1
2
3
2
4
5
3
6
5%
100

100
100
Payments occur annually, but compounding
occurs every 6 months.
 Cannot use normal annuity valuation
techniques.
Method 1:
Compound each cash flow
0
1
1
2
3
2
4
5
3
6
5%
100
100
100
110.25
121.55
331.80
FV3 = $100(1.05)4 + $100(1.05)2 + $100
FV3 = $331.80
Method 2:
Financial calculator
Find the EAR and treat as an annuity.
 EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.

INPUTS
OUTPUT
3
10.25
0
-100
N
I/YR
PV
PMT
FV
331.80
When is periodic rate used?
 iPER is
often useful if cash flows occur
several times in a year.
Inflation
Inflation - Rate at which prices as a whole are
increasing.
Nominal Interest Rate - Rate at which money
invested grows.
Real Interest Rate - Rate at which the
purchasing power of an investment increases.
48
Inflation
1+ nominal interest rate
1  real interest rate =
1+inflation rate
approximation formula
Real int. rate  nominal int. rate - inflation rate
49
Inflation
Example
If the interest rate on one year govt. bonds is 5.0% and
the inflation rate is 2.2%, what is the real interest rate?
1  real interest rate = 1 + .050
1 + .022
1  real interest rate = 1.027
real interest rate = .027 or 2.7%
Approximat ion = .050 - .022 = .028 or 2.8%
50
Inflation
Be consistent in how you handle
inflation!!
 Use nominal interest rates to discount
nominal cash flows.
 Use real interest rates to discount real
cash flows.
 You will get the same results, whether
you use nominal or real figures

51
Inflation
Example
You own a lease that will cost you $8,000 next year,
increasing at 3% a year (the forecasted inflation rate)
for 3 additional years (4 years total). If discount rates
are 10% what is the present value cost of the lease?
52
Inflation
Answer – Nominal figures
Year
1
2
3
4
Cash Flow
8000
8000x1.03 = 8240
8000x1.03 2 = 8487.20
8000x1.03 3 = 8741.82
PV @ 10%
8000
1.10  7272.73
8240
 6809.92
1.102
8487.20
 6376.56
1.103
8741.82
 5970.78
1.104
$26,429.99
53
Inflation
Answer – Real Figures
Year
1
2
3
4
8000
1.03
8240
1.032
8487.20
1.033
8741.82
1.034
Cash Flow
= 7766.99
= 7766.99
= 7766.99
= 7766.99
PV@6.7961%
7766.99
1.068  7272.73
7766.99
 6809.92
1.0682
7766.99
 6376.56
1.0683
7766.99
 5970.78
1.0684
= $26,429.99
54