Chapter 3:
Time Value of Money (TVM)
or: Discounted Cash Flow (DCF)
Professor Thomson
Fin 3014
Albert Einstein
• Professor Einstein was asked near the end
of his career what his greatest discovery
was. He replied . . .
2
Time Value of Money
• TVM is the foundation of the “House of Value”
that we are building this semester
• The fundamental equation of finance can be
stated as:
CF3
CF1
CF2
NPV  CF0 


 ...
1
2
3
(1  i ) (1  i ) (1  i)
3
Time Value of Money
Financial managers compare the marginal benefits
and marginal cost of investment projects.
Projects usually have a long-term horizon: timing of
benefits and costs matters.
Time-value of money concerned with adjusting for
different timing of benefits and costs.
4
The fundamental concept
• Value next period =
Value this period + interest earned
• Interest earned is the interest rate this
period times the Value deposited
Value
Time
5
Vt
Vt+1
---+----------------------------------+--t
t+1
Basic difference equation
• Vt+1 = Vt + it+1* Vt
Vt+1 = Value next period (period t+1)
Vt = Value this period
it+1 = Interest rate over period from t
to t+1
By algebra we can write:
• Vt+1 = Vt (1+ it+1)
6
See handwritten notes
• Note: The handwritten notes and
examples that follow go hand in hand.
7
Example 3.1
• You deposit $105 into a bank that pays
7% interest per year. How much will
you have after 1 year?
8
Example 3.2
• You deposit $100 into a bank that pays
5% interest per year. How much will
you have after 1 year?
9
Example 3.3
• You deposit $100 into a bank that pays
5% interest the first year, and 7%
interest the second year. How much
will you have after 2 years?
10
Example 3.4
• You deposit $100 into a bank that pays 5%
interest the first year, and 7% interest the
second year. How much have you earned in
interest over the two years?
• If you withdraw $5 after the first year, and
then withdraw your balance after the second
year, how much will you have earned in
interest over the two years?
11
Defn. Compound Interest
• Compound interest is earning interest on
your original deposit, plus earning
interest on interest
• Note: Interest on interest (and thus
compound interest) begins in the second
period.
12
Example 3.5
• You purchase a security for $100 that
earns 5% the first year, 7% the
second year, and loses 4% the third
year. What is the value of your
investment after 3 years?
13
Example 3.6
• You deposit $200 into a bank that pays
7% interest per year. How much will
you have after 4 years if you don’t
withdraw any?
14
Future Value (for a single sum with a
constant interest rate)
The value of a lump sum or stream of cash
payments at a future point in time:
FVN = PV x (1+i)N
Future Value
depends on:
Interest rate
Number of periods
Compounding interval
15
Future Value of $200
(4 Years, 7% Interest )
FV4 = $262.16
FV3 = $245.01
FV2 = $228.98
FV1 = $214
PV = $200
0
1
2
3
4
End of Year
Compounding: the process of earning
interest in each successive year
16
Compounding
Year 1:
FV1 = $214
Year 2:
FV2 = $228.98
Year 3:
FV3 = $245.01
Year 4:
17
FV4 = $262.16
• Earns 7% interest on initial $200
• FV1 = $200+$14 = $214
• Earn $14 interest again on $200 principal
• Earns $0.98 on previous year’s interest of
$14: $14 x 7% = $0.98
• FV2 = $214+$14+$0.98 = $228.98
• Earn $14 interest again on $200 principal
• Earns $2.03 on previous years’ interest of
$28.98: $28.98 x 7% = $2.03
• FV3 = $228.98+$14+$2.03 = $245.01
• Earn $14 interest again on $200 principal
• Earns $3.15 on previous years’ interest of
$45.01: $45.01 x 7% = $3.15
• FV4 = $245.01+$14+$3.15 = $262.16
Calculator hints
• Clear the calculator before new problems
(Use the  C ALL)
• Make sure:
– The desired number of decimal places are
displayed
• Set using  DISP followed by entering a digit
– You have the correct payments (periods) per
year
• Set by typing a number then press 
P/YR
18
• Check by holding down  C ALL
Calculator hints (continued)
BEGIN indicator is not displayed, unless you
are told this problem has beginning of
period cash flows
– Set using  BEG/END
If you have a comma where you should
have a decimal point (European notation)
then toggle to decimal by:
– Toggle using  ./,
19
Notation when using Calculator
•
•
•
•
•
•
•
20
P/YR = 1 (indicate the periods per year)
FV(PV= – 200, N= 4, I/YR=7) = 262.16
Order of inputs does not matter
Negative sign for PV indicates a cash outflow
N = number of periods
I/YR = stated annual interest rate
The last button one pushes is what you want to
solve for – in this case FV.
The Power of Compound Interest
Note: If you are given any
point on one of these
curves, you should be able
to determine any other point
41
36
31
20%
26
21
15%
16
11
10%
6
5%
0%
1
1
21
3
5
7
9
11
13
Periods
15
17
19
21
23
25
What if one of Jesus’ disciples was
an investor, and you were his heir?
• A newly discovered gospel suggests a
disciple put $1 in the First Bank of Galilee
which promised to pay 5% per year. How
much is that deposit worth today?
(Assume deposit was in year 25 AD, so N
= 2006 – 25 = 1981 yr.)
• FV(PV=-1, I/YR=5, N=1981) =
= 9.46 * 1041 or approx. 1042
22
How much is that?
• Is that more than Bill Gates has?
• Bill Gates has give or take $50 Billion (in
orders of magnitude say 1011)
• World population:6,520,566,380
See http://www.census.gov/main/www/popclock.html
• If you divided the account among the
world population we would each have
about: 1032 which would make Bill Gates
poor.
• Do you get Einstein’s point?
23
Example 3.7
• You deposit $250 into a bank that pays
7% per year. At the end of 3 years, you
withdraw $100, and leave the rest in the
bank. How much will you have in your
account after 8 years?
24
What about solving for PV?
• Often what we want to know is the value
in today's framework of money we will
receive in the future
• We call this process discounting
• We use discounting to compute the
Present Value (PV)
• Use some algebra: Take the future value
equation, and divide through by (1+i)N
25
Present Value
Today's value of a lump sum or stream of cash
payments received at a future point in time:
FVN  PV  1  i 
N
PV 
26
FVN
(1  i )
N
Example 3.8
• A savings bond will pay you $200 on your
27th birthday, which is 4 years away.
Assuming a discount rate of 7%, what is
the present value of this payment?
27
By financial calculator
• P/YR = 1
• PV(FV=200, N=4, I/YR=7) = – 152.58
28
Present Value of $200
(4 Years, 7% Interest )
Discounting
0
1
FV1 = $200
2
FV2 = $200
3
FV3 = $200
4
FV4 = $200
End of Year
PV = $186.92
PV = $174.69
PV = $163.26
PV = $152.58
29
Discounting: the process of converting a future
cash flow into a present value
The Power of High Discount Rates
1.00
0%
0.75
0.5
5%
0.25
30
10%
15%
20%
0 2 4 6 8 10 12 14 16 18 20 22 24
Periods
Example 3.9
• Tuition and fees for a semester was about
$1800 five years ago, and now is about
3100. What is the annual rate of
increase?
31
32
Example 3.10 Stradivarius
Violin Example
• According the the April 3, 1998 Wall Street
Journal, musician George Kress sold his
Stradivarius violin for $1.58 Million. He
had purchased said violin in 1958 for
$24,000. What rate of return did he earn
on this violin.
33
Example 3.11 Or, invest in
Stock Market
• A popular stock market index had a value
of 25.3 in 1958, and 2347 in 1998. If
George had simple put his $24000 into
this index, what would his investment be
worth today? What rate of return would
he have earned? Would this have been
better than buying the Stradivarius violin?
34
From WSJ April 12, 2007 (P D7)
• Record price for Strad (2006): $3,544,000
• The “Solomon - ex Lambert” sold in April
2007 for $2,700,000 (had been purchased
for about $43,000 in 1972 (noted to not
be one of the best Strads). From 1972 to
the end of 1996, the overall stock market
went up about 46.41 times (including
reinvestment of all dividends).
• What rate of return on the Strad and stock
market was realized over this period?
35
Example 3.12
• UTSA tuition has been increasing at about
10% each year. How long will it take for
tuition to double? (Will you graduate
before it doubles?)
36
The rule of 72
• To find the period for an investment, growing at
rate i, takes to double, use the rule of 72
72
PeriodToDo uble 
InterestRa te%
Example: Let i = 10% per year
37
72
 7.2 years
10
Illustration of Rule of 72
25
Period to Double .
20
15
Rule of 72
Exact
10
5
0
3
4
5
6
7
8
9
10
Interest Rate %
38
11
12
13
14
15
Dealing with Multiple Cash Flows
•
There are two basic options for dealing
with multiple cash flows:
1. Treat each CF individually, and sum up
the results
– Must use this approach if the CF’s are not
equal in amount or are irregularly spaced
2. Create new formulas to use when the
CF’s are equal and regularly spaced
39
Example 3.13 – Non equal CF’s
What is the present value (using a 5%
discount rate) of the following CF’s?
Year
CF
1
2
3
4
100
50
0
325
What is the PV using a 10% rate?
40
Solving using the CF and NPV keys
• The calculator has the following programmed:
CF3
CF1
CF2
NPV  CF0 


 ...
1
2
3
(1  i ) (1  i ) (1  i)
One accesses this function by entering a series
of Cash Flows, with the first cash flow assumed
to be a period 0. After all cash flows are
entered, enter the I/YR and press the Shift NPV
sequence to display the NPV
41
NPV indicates Net Present Value, where net usually
means to subtract off the initial CF
Calculator Solution
• P/YR = 1
• Enter the CF’s
t
CF
0
0
1 100
2
50
3
0
4 325
•Type 5, press I/YR
42
•Press  NPV
Example 3.14 – FV of Non equal CF’s
What is the future value (using a 5%
discount rate) of the following CF’s?
Year
CF
1
2
3
4
100
250
550
325
What is the FV using a 10% rate?
43
Regularly spaced equal payments
• Perpetuity – A series of equally spaced,
equal amount cash flows that continue in
perpetuity (i.e. forever). The first
payment is assumed to be one period
from now.
• Annuity – A series of equally spaced,
equal amount cash flows that end at time
period N.
44
Creating a Perpetuity
If you deposit $100 in a bank that pays 5% per
year, how much interest will you have earned
after one year?
Answer: $5
If you withdraw the $5 and and reinvest your
$100, how much interest will you have after
one additional year? $5.
Note: You could do this forever. In other
words, you can create a payment of $5 to
yourself, every year, forever, by depositing $100
into a bank that pays 5% per year.
45
You could write this statement down as a
formula that would be:
PMT = i * Original deposit
The original deposit is made today, so it
is a Present Value, so we can write:
PMT = i * PV
Using algebra we can write this as a
Present Value formula:
PV = PMT/i
46
This is the present value of a perpetuity.
NOTE: The first PMT is at period 1, (i.e.
it is one period from today).
Example 3.15 – A perpetuity
• KindaCheep Life Insurance Co. is offering
you an investment policy that will pay you
and your heirs $10 per year forever. If
your required return on this investment is
5%, how much will you pay for the policy?
47
Example 3.16:
Delayed perpetuity
• EvenCheeper Life Insurance Co. is offering
you an investment policy that will pay you
and your heirs $10 per year forever, with
your first payment 4 years from today. If
your required return on this investment is
5%, how much will you pay for the policy?
48
Example 3.17: PV of Annuity
• What is the present value of receiving $10
per year for 3 years if the discount rate is
5%?
• You could discount term by term using of
PV of a single cash flow formula and add
up the individual present values
• However, this can be computed as:
• 200.00 – 172.77 = 27.23
• Why?
49
Diagram to derive annuity formula
50
Formula for PV of Annuity
• It’s the present value of a perpetuity minus
the present value of a delayed perpetuity
Pmt Pmt / i
PV 

N
i
(1  i )
51
Solving by Financial Calculator
• We use a “new” key, the PMT key
P/YR=1
PV(PMT=-10, I/YR=5, N=3) = 27.23
52
FV of an Annuity
• To compute the FV of an Annuity, you can
simple multiply the PV by (1+i)N
• Or, use your financial calculator where the
last key you press is FV
• What about the FV of a perpetuity?
– It is undefined as you are accumulating
forever
53
Example 3.18: FV of Annuity
Saving for Retirement
• What is the FV of investing $3600 per year for
40 years if you invest at 6%, if your first
investment is 1 year from today?
P/YR = 1
FV(PMT=-3600, N=40, I/YR=6) = $557,143.08
• Is this enough to retire on?
54
Habit 2: Begin with the end in mind
• How much money do you want to spend
in retirement? How long do you want to
have income in retirement?
• Example: I want to have an income of
$36000 per year for 25 years. At 6%,
how much money do I need in my
retirement fund?
P/YR=1
PV(PMT=36000, N=25, I/YR=6) = -460,200.82
55
If you die after 25 years, how much
do your heirs get?
• The balance in your retirement account will
always be the present value of the remaining
payments. If you have $557,143.08 in your
account, and if you withdraw 36000 in each
year, we can compute the number of years you
can get such payments (still assuming a 6%
interest rate).
P/YR = 1
N(PV=-557143.08, PMT=36000, I/YR=6)=45.29
56
How large is the inheritance
• Your heirs could continue to withdraw
money for 20.29 years.
P/YR = 1
PV(PMT=36000, N=20.29, I/YR=6)
= 416,051.92
• This number is quite high, because during
retirement your were mainly living off your
interest
• First year interest amount in retirement was:
$557,143.08*0.06 = 33,428.58
57
• Which perhaps explains why Mark
Twain said, “Thrift if a virtue –
especially in ones ancestors”
58
Future Value and Present Value of
an Ordinary Annuity
Compounding
Future
Value
$1,000
0
1
$1,000
2
$1,000
$1,000
3
End of Year
Present
Value
59
Discounting
4
$1,000
5
Future Value of Ordinary Annuity
(End of 5 Years, 5.5% Interest Rate)
$1,238.82
$1,174.24
$1,113.02
$1,055.00
$1,000.00
0
$1,000
$1,000
$1,000
1
2
3
$1,000
End of Year
4
$1,000
5
(1  r )  1
FV  PMT 
 $5,581.08
r
n
How is annuity due different ?
60
Future Value of Annuity Due
(End of 5 Years, 5.5% Interest Rate)
$1,306.96
$1,238.82
$1,174.24
$1,113.02
$1,055.00
$1,000
0
$1,000
1
$1,000
2
$1,000
$1,000
3
4
End of Year
5
(1  r ) n  1
FV  PMT 
 1  r   $5,888.04
r
61
Annuity due: payments occur at the beginning of each
period
Present Value of Ordinary Annuity
(5 Years, 5.5% Interest Rate)
0
1
$1,000
2
$1,000
3
$1,000
4
$1,000
5
$1,000
End of Year
$947.87
$898.45
$851.61
$807.22
$765.13
62
PMT 
1 
PV 
 1 
 $4,270.28
n 
r
 (1  r ) 
Present Value of Annuity Due
(5 Years, 5.5% Interest Rate)
0
$1,000
1
$1,000
2
$1,000
3
$1,000
4
5
$1,000
End of Year
$947.87
$898.45
$851.61
$807.22
PMT 
1 
PV 
 1 
 1  r   $4,505.15
n 
r
 (1  r ) 
63
Example 3.18 revised: FV of
Annuity - Saving for Retirement
• Original Version: What is the FV of investing $3600 per
year for 40 years if you invest at 6% if your first
investment is 1 year from today?
64
• Revised version: What is the FV of investing
$300 per month for 480 month if you invest at
0.5% if your first investment is 1 month from
today?
P/YR = 1
FV(PMT=-300, N=480, I/YR=0.5) = $597,447.22
Why is this about $40,300 more than the original?
 Higher interest rate (.5% per month is more
than 6% per year because compounding starts
in month 2 rather than year 2. Also, higher
annual payment – Why?
The Power of Compound Interest
Note: If you are given any
point on one of these
curves, you should be able
to determine any other point
41
36
31
20%
26
21
15%
16
11
10%
6
5%
0%
1
1
65
3
5
7
9
11
13
Periods
15
17
19
21
23
25
Example 3.19 Interest rate for
period other than 1 year.
If you invest one dollar at 5% per year, how
much interest will you have earned after:
• A) 2 years (This is the 2 year interest rate
for a 5% per year rate)
• B) 2 months (This is the 2 month interest
rate for a 5% per year rate)
66
Example 3.20
Interest Earned After 1 Year: Invest $1:
• If you earn 0.11 interest after 3 years,
how much did you earn after 1 year?
• This takes a interest rate earned after
some period and converts to the interest
rate earned each year. This is called the
EAR – Effective Annual Rate (or EAY –
Effective Annual Yield).
67
Example 3.21
Interest Earned After 1 Year: Invest $1:
• If you earn 0.03 interest after 2 months,
how much would you earn in 1 year
• Once again, this is an example of
computing an EAR.
68
Effective Annual Rate: EAR
• The EAR, takes an interest rate earned
over any period of time, and computes the
interest you would earn after one year.
• The EAR is the amount of interest you
earn on an investment in one year, per
one dollar invested.
• EAR = (1+i)m-1, where m=number of
periods per year. It is the FVIF for a one
year period, minus your original $1
investment
69
Compounding Intervals
m compounding periods per year
APR 

FVN  PV  1 

m 

mYr
Note: APR is the stated interest rate. APR/m is
the rate used for compounding, i
70
The more frequent the compounding period,
for a given stated rate, the larger the FV!
Compounding More Frequently Than
Annually
FV at end of 2 years of $125,000 deposited at 5% stated
interest (APR)
– For semiannual compounding, m equals 2:
 0.05 
FV2  $125,000  1 

2 

2 2
 $137,976.61
– For quarterly compounding, m equals 4:
71
 0.05 
FV2  $125,000  1 

4 

4 2
 $138,060.76
Compounding More Frequently Than
Annually
FV at end of 2 years of $125,000 deposited at 5% stated
interest (APR)
– For monthly compounding, m equals 12:
122
 0.05 
FV2  $125,000  1 

12 

 $138,117.67
– For daily compounding, m equals 365:
72
 0.05 
FV2  $125,000  1 

365 

3652
 $138,145.42
Continuous Compounding
• In extreme case, interest compounded
continuously:
FVN = PV x (e r x Yr)
FV at end of 2 years of $125,000 at 5 % annual interest,
compounded continuously:
FV2  $125,000  e
73
0.05 2
 $138,146.365
Handling multiple periods per year
using the HP10BII
FV at end of 2 years of $125,000 deposited at 5% interest
Annual: P/YR=1
FV(PV=-125000, I/YR=5, N=2) = 137,812.50
SemiAnnual: P/YR=2
FV(PV=-125000, I/YR=5, N=4) = 137,976.61
Monthly: P/YR=12
FV(PV=-125000, I/YR=5, N=24) = 138,117.67
Daily: P/YR=365
FV(PV=-125000, I/YR=5, N=730) = 138,145.42
74
How the HP10BII calculator treats
multiple periods per year
• The interest rate the HP10BII calculator uses for
performing its calculations is : I / YR
P / YR
Example: What is the future value of $125 after 2 years at
5% compounded monthly?
P/YR = 12, FV(PV=-125, N=24, I/YR=5) = 138.12
N
24
I / YR 

 0.05 
FV  PV 1 
  1251 
  138.12
12 
 P / YR 

Note: To solve using years, must use the EAR as the
interest rate: FV = 125*(1.052379)2 = 138.12
75
The Stated Rate versus the Effective
Rate Rate per period = r/m
Stated rate: the contractual annual rate charged
by lender or promised by borrower
This is called the APR: Annual Percentage Rate
APR = (r/m) * m = r
Effective rate: the annual rate actually paid or
earned
m
APR 

EAR  1 
 1
m 

76
The Stated Rate versus the
Effective Rate
• FV of $100 at end of 1 year, invested at 8%
stated annual interest (APR), compounded:
– Annually:
FV = $100 (1.08)1 = $108.00
– Semiannually: FV = $100 (1.04)2 = $108.16
– Quarterly:
FV = $100 (1.02)4 = $108.24
Stated rate (APR) of 8% does not change.
What about the effective rate?
77
Effective Rates: Always Greater
Than or Equal to Stated Rates
• For annual compounding, effective = stated
1
0.08 

EAR  1 
  1  8.00%
1 

• For semiannual compounding
2
0.08 

EAR  1 
  1  8.16%
2 

• For quarterly compounding
4
78
0.08 

EAR  1 
  1  8.24%
4 

Computing the EAR using the
HP10BII
The key for computing the EAR is  EFF%
To compute the EAR for 8% compounded
quarterly
P/YR=4
 EFF% (I/YR=8) = 8.2432
79
Annual vs. Quarterly Compounding
100.00
108.00
+-----------------------------------------------+
Yr 0
i=8%
Yr 1
100.00 102.00 104.04 106.12 108.24
+-----------+----------+----------+----------+
Q0
Q1
Q2
Q3
Q4
i=2%
80
i=2%
i=2%
i=2%
Example 3.18 revised again: FV of
Annuity - Saving for Retirement
• Original Version: What is the FV of investing $3600 per year for 40
years if you invest at 6% if your first investment is 1 year from
today? $557,143.08
• Revised version: What is the FV of investing $300 per
month for 480 month if you invest at 0.5% if your first
investment is 1 month from today? $597,447.22
• What if the original version was at the EAR of the
revised version, and had the same yearly investment?
• EAR = (1.005)12 – 1 = 0.061678 = 6.1678%
• $300 per month, invested a .5% per month for one year
P/YR=1 FV(PMT=-300,N=12, I/YR=.5) = 3,700.67
Invest this amount for 40 years: P/YR = 1
FV(PMT=-3700.67, N=40, I/YR=6.1678) = $597,450.27
81
Reminder
• Interest rate and time period must always be in
agreement!!!!
• 1% per month is not the same as 12% per year
• 1% per month is equivalent to 12.6825% (the
EAR) per year because of compounding.
• 12% compounded monthly is a common way to
say that you are earning 1% per month. In this
case, 12% is the APR, or stated rate, but you
earn (or pay) at the rate of 12.6825%.
82
PV of $100 received one year from today,
discounted at 12% compounded monthly
Two approaches:
1. Use the monthly rate and discount for 12
months.
100/(1.01)12 = 88.74, or by FinCalc
P/YR=1, PV(N=12,I/YR=1,FV=100), or
P/YR=12, PV(N=12,I/YR=12,FV=100)
2. Use the correct yearly discount rate, the EAR,
and discount for 1 year.
100/(1.126825) = 88.74, or by FinCalc
P/YR=1, PV(N=1,I/YR=12.6825,FV=100)
83
Restating our Formulas
• FV = PV(1+APR/m)N
• Where:
APR = annual interest rate as APR (I/YR)
m = number of periods per year (P/YR)
N = Number of periods
N = Number of years x m
• FV =
(YR*m)
PV(1+APR/m)
i as used in our formulas is APR/m
84
Example 3.22 Financing a car
• You have $2500 to put down on a new car
that you have bargained to $24962
including Tax, Title, License, and all fees.
Two financing options are:
• 36 months at 6.9%
• 72 months at 7.7%
• What is your monthly loan payment for
each option?
• How much do you pay into interest for
each option?
85
Example 3.23 Cash back or lower
interest rate
• You require a loan of $20,000 on your car
purchase and can either have a 1.8%
finance rate for 48 months or $1000 cash
back. If you choose the cash back, it will
cost you 5.7% for 48 months? Which is
the best alternative? Why?
86
87
88
Example 3.24 Getting rid of
plastic hubcaps
• Kristy Turney got rid of her plastic
hubcaps just 4 days after buying her “new
used car.” She received a new set of 1800
wheels by agreeing to pay $57 per week
for 52 weeks.
• How much interest will Ms Turney pay?
• What is the APR and the EAR for this
“loan”?
89
Example 3.25 Social Security
versus personal investment account
You are currently age 22 and will invest
12.3% of your (constant) gross income
through age 67 then withdraw through
your life expectancy
Social Security Actuarial Assumptions
– Age 22 Male will live to 74
– Age 22 Female will live to 79
• How much of your salary will your
investment account replace?
90
Social Security or Invest 12.3%
Salary Replacement Factor
6
5
Social Security Actuarial Assumptions
Age 22 Male will live to 74
Age 22 Female will live to 79
Contribute from Age 22 through Age 67
4
3
2
1
Male Factor
Female Factor
0
3%
91
4%
5%
Rate of Return
6%
7%
Social Security Benefits Retire 2045
80%
Current Benefits
Insolvent
70%
Progressive Index
60%
50%
40%
30%
20%
10%
0%
16470
36600
58560
Wages in 2005
92
90000
Support for Social Security Changes
Under 55
55 and Older
70%
60%
50%
40%
30%
20%
10%
0%
Raise
Limit
Allow
payroll tax benefits for private
cap
wealthy
accounts
93
Increase
Payroll tax
rate
Slow
growth of
benefits
Raise
retirement
age
Example 3.26.
• Your friendly loan shark makes you a “2
for 4 or I knock at your door” offer, which
means he will loan you $200 today, but
you must pay him back $400 in one
month, or else you get the knock at your
door. What is the APR and EAR of this
financial transaction?
94
Example 3.27
• You have $100 to invest and the following
options are available. Invest at 5% per
year for three years, or invest at 6% for
two years. For you to have an equal
amount in your account after three years,
at what rate would you have to invest in
the third year, if you chose the two year
investment?
95
Example 3.28 Skipping Perpetuity
• What is the present value of receiving
$100 every 3 months if the discount rate
is 12% compounded monthly?
96
Example 3.29 Skipping Perpetuity
• What is the present value of receiving
$100 one year from now, $200, two years
from now, $100 three years from now,
$200 four years from now, and so on, in
perpetuity if the stated rate is 5%
compounded annually?
97
Example 3.30 – Non equal CF’s
(Example 3.13, revised). What is the
present value (using a 5%, compounded
quarterly discount rate) of the following
CF’s?
Year
CF
98
1
2
3
4
100
50
0
325
Present Value of Growing
Perpetuity
CF1
PV0 
rg
rg
0
1
$1,000
$1,000
2
3
$1,000(1+0.02)1 $1,000(1+0.02)2
$1,020
$1,040.4
4
$1,000(1+0.02)3 …
$1,061.2
Growing Perpetuity
CF1 = $1,000
r = 7% per year
99
g = 2% per year
$1,000
PV0 
 $20,000
0.07  0.02
Example 3.31
• A growing perpetuity. Even Better
Assurance Company offers you a
perpetuity whose payment will grow at
3% every year. The first payment of
$1500 will be one year from now. For a
9% discount rate, what is the present
value of this growing perpetuity?
100
Much Of Finance Involves
Finding Future And (Especially)
Present Values
 Central to all financial valuation techniques

Techniques used by investors and firms alike