Chapter
Six
Discounted Cash
Flow Valuation
© 2003 The McGraw-Hill Companies, Inc. All rights reserved.
6.1
Key Concepts and Skills
 Be able to compute the future value of
multiple cash flows
 Be able to compute the present value of
multiple cash flows
 Be able to compute loan payments
 Be able to find the interest rate on a loan
 Understand how loans are amortized or paid
off
 Understand how interest rates are quoted
6.2
Chapter Outline
 Future and Present Values of Multiple Cash
Flows
 Valuing Level Cash Flows: Annuities and
Perpetuities
 Comparing Rates: The Effect of Compounding
 Loan Types and Loan Amortization
6.3
Multiple Cash Flows 6.1 – FV Example 1
 You currently have $7,000 in a bank account earning 8%
interest. You think you will be able to deposit an additional
$4,000 at the end of each of the next three years. How much
will you have in three years?
0
1
2
7,000
4,000
4,000
FV  PV 1  r 
t
3
4,000.00
4,320.00
4,665.60
8,817.98
$21,803.58
6.4
Multiple Cash Flows – FV Example 2
 Suppose you invest $500 in a mutual fund today and $600 in
one year. If the fund pays 9% annually, how much will you
have in two years?
0
1
500
600
2
654.00
594.05
$1,248.05
9%
6.5
Multiple Cash Flows – FV Example 2 Continued
 Using the cash flows from the previous page, how much will
you have at the end of 5 years, if you were to make no further
deposits?
0
1
500
600
2
3
4
5
9%
846.95
769.31
$ 1,616.26
6.6
Multiple Cash Flows – FV Example 3
 Suppose you plan to deposit $100 into an account in one year
and $300 into the account in three years. How much will be
in the account in five years if the interest rate is 8%?
0
1
100
2
3
4
5
8%
300
349.92
136.05
$ 485.97
6.7
Multiple Cash Flows – PV Example 1
 You are offered an investment that will pay you $200 in one
year, $400 the next year, $600 the year after, and $800 at the
end of the following year. You can earn 12% on similar
investments. How much is this investment worth today?
0
1
2
3
4
200
400
600
800
178.57
318.88
427.07
508.41
1,432.93
PV 
FV
1  r 
t
12%
6.8
Multiple Cash Flows – PV Example 2
 You are considering an investment that will pay you $1000 in
one year, $2000 in two years and $3000 in three years. If you
want to earn 10% on your money, how much would you be
willing to pay?
Answer:
$4,815.93
6.9
Multiple Cash Flows Using a Spreadsheet
 You can use the PV or FV functions in Excel to find the
present value or future value of a set of cash flows
 Setting the data up is half the battle – if it is set up properly,
then you can just copy the formulas
 Click on the Excel icon for an example
6.10
Uneven Cash Flows – Using the Calculator
 Another way to use the financial calculator for uneven cash
flows is to use the cash flow keys
 HP 10BII
 Enter the first cash flow, using the +/- keys to indicate
outflows
 Press CF
 Enter the second cash flow
 Press CF
 Enter the interest rate as I/Y
 Use the 2nd function, then the NPV key to compute the
present value
 To clear this function, use 2nd function, Clear All
6.11
Decisions, Decisions
 Your broker calls you and tells you that he has this great
investment opportunity. If you invest $100 today, you will
receive $40 in one year and $75 in two years. If you require a
15% return on investments of this risk, should you take the
investment?
 Use the CF keys to compute the value of the investment
100 +/CF
Note: NPV (Net Present Value) is the
addition to, or the destruction of, wealth
40
CF
that occurs from undertaking an
75
CF
investment. If NPV is positive, you
15
I/Y
create wealth. If NPV is negative, you
destroy wealth. Therefore, the decision
2nd NPV
-8.51
Reject: NPV < 0
rule is to accept only projects with an
NPV > 0
6.12
Quick Quiz – Part I
 Suppose you are looking at the following possible cash flows:




Year 1 CF = $100;
Years 2 and 3 CFs = $200;
Years 4 and 5 CFs = $300.
The required discount rate is 7%
 What is the value of the cash flows at year 5?
 What is the value of the cash flows today?
 What is the value of the cash flows at year 3?
6.13
Annuities and Perpetuities 6.2
 Annuity – finite series of equal payments that occur
at regular intervals
 If the first payment occurs at the end of the period, it is
called an ordinary annuity
 If the first payment occurs at the beginning of the period, it
is called an annuity due
 Perpetuity – infinite series of equal payments
6.14
Annuities – Basic Formulas
 Annuities:
PVOrdinary
Annuity
FVOrdinary
Annuity
 1 -  1+ r   t
=C
r


  1+ r t  1 
=C

r




Where:
PV = Present Value
FV = Future Value
C = the equal periodic cash flow
R = interest or discount rate
t = the number of time periods




6.15
Ordinary Annuities Versus Annuities Due
 Ordinary Annuity – first payment occurs at
time period 1
 Annuity Due – first payment occurs at time
period 0
 1 -  1+ r)  t 
PVAnnuity = C 
 1  r 
Due
FVAnnuity
Due


r


  1+ r t  1 
=C
 1  r 
r




6.16
Annuities and the Calculator
 To solve any annuity type problem using the function
keys, you must use the PMT key
 Only use the PMT key when each cash flow is exactly the
same size and occurs at regular intervals
 Ordinary annuity versus annuity due
 To switch the calculator between an ordinary annuity and
an annuity due, enter 2nd BGN (HP 10BII)
 If you see “BGN” or “Begin” in the display of your
calculator, you have it set for an annuity due
 Most problems are ordinary annuities
6.17
Annuity – Example 1
 After carefully going over your budget, you have determined
that you can afford to pay $632 per month towards a new
sports car. Your bank will lend to you at 1% per month for 48
months. How much can you borrow?
 Formula Approach
Calculator Approach
 1 -  1+ r -t 
632
PMT
PV = Pmt 

r


0
FV
48
N
 1 -  1.01-48 
= 632 

1
I/Y
.01


PV
$23,999.54
= 23,999.54
6.18
Annuity – Sweepstakes Example
 Suppose you win the Publishers Clearinghouse $10 million
sweepstakes. The money is paid in equal annual
installments of $333,333.33 over 30 years. If the
appropriate discount rate is 5%, how much is the
sweepstakes actually worth today?
Formula Approach
1  1  r  t 
PVAnnuity  PMT 

r


1  1.0530 
 333,333.33

.
05


 $5,124,150.29
Calculator Approach
333,333.33
PMT
0
FV
30
N
5
I
PV
$5,124,150.29
6.19
Annuities on the Spreadsheet - Example
 The present value and future value formulas in
a spreadsheet include a place for annuity
payments
 Click on the Excel icon to see an example
6.20
Quick Quiz – Part II
 You know the payment amount for a loan and
you want to know how much was borrowed.
Do you compute a present value or a future
value?
 You want to receive $5,000 per month in
retirement. If you expect to earn 9 per year,
compounded monthly and you expect to need
the income for 25 years, how much do you
need to have in your account at retirement?
6.21
Finding the Payment
 Suppose you want to borrow $20,000 for a new car. You can
borrow at 8% per year, compounded monthly (8%/12 =
0.66667% per month). If you take a 4 year loan, what is your
monthly payment?
Formula Approach
1  1  r t 
PVAnnuity  PMT 

r


  .08   4 x12 

1  1 

12



20,000  PMT 
.08




12


PMT  $488.26
Calculator Approach
20,000
PV
0
FV
4 x 12 =
N
8 ÷ 12 =
I
PMT
$488.26
6.22
Finding the Payment on a Spreadsheet
 Another TVM formula that can be found in a
spreadsheet is the payment formula
 PMT(rate,nper,pv,fv)
 The same sign convention holds as for the PV and
FV formulas
 Click on the Excel icon for an example
6.23
Finding the Number of Payments – Example 1
 You ran a little short on your February vacation, so
you put $1,000 on your credit card. You can only
afford to make the minimum payment of $20 per
month. The interest rate on the credit card is 1.5%
per month. How long will you need to pay off the
$1,000?
6.24
Finding the Number of Payments – Example 1
 Formula Approach
 Start with the basic annuity formula and solve for the
unknown exponent t, using logs.
1  1  r t 
PVAnnuity  PMT 

r


1  1.015t 
1,000  20

.
015


1  1.015t 
50  

.
015


Calculator Approach
1,000
PV
0
FV
20 +/PMT
1.5
I
N
93.11 months
0.75  1  1.015
t
0.25  1.015
ln 0.25  t ln 1.015
t  93.11 months or 7.76 years
t
6.25
Finding the Number of Payments – Example 2
 Suppose you borrow $2000 at 5% and you are going to make
annual payments of $734.42. How long before you pay off
the loan?
1  1  r t 
PVAnnuity  PMT 

r


1  1.05t 
2,000  734.42

.
05


1  1.05t 
2.72  

.
05


0.1362  1  1.05
t
0.8638  1.05
ln 0.8638  t ln 1.05
t  3.0 years
t
Calculator Approach
2,000
PV
0
FV
734.42 +/PMT
5
I
N
3 years
6.26
Finding the Rate On the Financial Calculator
 Suppose you borrow $10,000 from your parents to buy a car.
You agree to pay $207.58 per month for 60 months. What is
the monthly interest rate?
Calculator Approach
10,000
PV
0
FV
207.58 +/PMT
60
N
I
0.7499% per month
6.27
Annuity – Finding the Rate Without a Financial Calculator
 Trial and Error Process
 Choose an interest rate and compute the PV of the
payments based on this rate
 Compare the computed PV with the actual loan
amount
 If the computed PV > loan amount, then the interest
rate is too low
 If the computed PV < loan amount, then the interest
rate is too high
 Adjust the rate and repeat the process until the
computed PV and the loan amount are equal
6.28
Quick Quiz – Part III
 You want to receive $5000 per month for the next 5 years.
How much would you need to deposit today if you can earn
0.75% per month?
 What monthly rate would you need to earn if you only have
$200,000 to deposit?
 Suppose you have $200,000 to deposit and can earn 0.75% per
month.
 How many months could you receive the $5000 payment?
 How much could you receive every month for 5 years?
6.29
Future Values for Annuities – Example 1
 Suppose you begin saving for your retirement by depositing
$2000 per year in an RRSP. If the interest rate is 7.5%, how
much will you have in 40 years?
Formula Approach
 1  r t  1
FVAnnuity  PMT 

r


 1.07540  1
 2,000 

.
075


 454,513.04
Calculator Approach
2,000
PMT
0
PV
40
N
7.5
I
FV
$454,513.04
6.30
Annuity Due – Example 1
 You are saving for a new house and you put $10,000 per
year in an account paying 8% compounded annually. The
first payment is made today. How much will you have at the
end of 3 years?
Formula Approach
 1  r t  1
FVAnnuity  PMT 
 1  r 
r
Due


 1.083  1
 10,000 
 1.08
 .08 
 $35,061.12
Calculator Approach
2nd BGN
10,000
PMT
0
PV
3
N
8
I
FV
$35,061.12
6.31
Perpetuity
 Perpetuity: A stream of cash flows that continues
forever
PVPerpetuity =
C1
r
Where:
PV = Present Value
C = the equal periodic cash flow, starting at time period 1
r = discount rate
6.32
Perpetuity – Example 1
 The Home Bank of Canada want to sell
preferred stock at $100 per share. A very
similar issue of preferred stock already
outstanding has a price of $40 per share and
offers a dividend of $1 every quarter. What
dividend would the Home Bank have to offer
if its preferred stock is going to sell?
6.33
Perpetuity – Example 1 continued
 First, find the required return for the comparable issue:
C1
r
$1.00
40 
r
r  2.5% per quarter
PVPer 
 Then, using the required return found above, find the
dividend for new preferred issue:
C1
r
C1
100 
0.025
C1  $2.50 per quarter
PVPer 
6.34
Growing Perpetuity
 The perpetuities discussed so far are annuities with constant
payments
 Growing perpetuities have cash flows that grow at a
constant rate and continue forever
 Growing perpetuity formula:
C1
PVPerpetuity 
rg
W ithGrowth
6.35
Growing Perpetuity – Example 1
 Hoffstein Corporation is expected to pay a dividend of $3
per share next year. Investors anticipate that the annual
dividend will rise by 6% per year forever. The required
rate of return is 11%. What is the price of the stock today?
C1
PV 
rg
$3.00

0.11  0.06
 $60.00
6.36
Growing Annuity
 Growing annuities have a finite number of cash
flows, which grow at a constant rate
 The formula for a growing annuity is:
T

C1
1 g  
PV 
 
1  
r  g   1  r  
6.37
Growing Annuity – Example 1
 Gilles Lebouder has just been offered a job paying $50,000
at the end of his first year. He anticipates his salary will then
increase by 5% a year until his retirement in 40 years. Given
an interest rate of 8%, what is the present value of his
lifetime salary?
t
C1   1  g  
PV 

1  
r  g   1  r  
40
$50,000   1.05  

 
1  
0.08  0.05   1.08  
 $1,126,571
6.38
Quick Quiz – Part IV
 You want to have $1 million to use for retirement in 35 years.
If you can earn 1% per month, how much do you need to
deposit on a monthly basis if the first payment is made in one
month?
 What if the first payment is made today?
 You are considering preferred stock that pays a quarterly
dividend of $1.50. If your desired return is 3% per quarter,
how much would you be willing to pay?
6.39
Work the Web Example
 Another online financial calculator can be found at
MoneyChimp
 Click on the web surfer and work the following
example
 Choose calculator and then annuity
 You just inherited $5 million. If you can earn 6% on your
money, how much can you withdraw each year for the next
40 years?
 Datachimp assumes an annuity due!!!
 Payment = $313,497.81
6.40
Effective Annual Rate (EAR) 6.3
 The Effective Annual Interest Rate is the actual rate
paid (or received) after accounting for compounding
that occurs during the year
 For example, 10% compounded twice a year has an
EAR of 10.25%. This means that 10%, compounded
twice a year, is identical to 10.25%, compounded
annually.
 If you want to compare two alternative investments
with different compounding periods, you need to
compute the EAR for both investments and then
compare the EAR’s.
6.41
Annual Percentage Rate (Nominal Rate)
 This is the annual rate that is quoted by law
 By definition, the nominal rate or APR =
period rate times the number of periods per
year
 Consequently, to get the period rate we
rearrange the APR equation:
 Period rate = APR / number of periods per year
 You should NEVER divide the effective rate
by the number of periods per year – it will
NOT give you the period rate
6.42
Computing APRs (Nominal Interest Rate)
 What is the APR if the monthly rate is .5%?
 .5(12) = 6%
 What is the APR if the semiannual rate is .5%?
 .5(2) = 1%
 What is the monthly rate if the APR is 12%
with monthly compounding?
 12 / 12 = 1%
6.43
Things to Remember
 You ALWAYS need to make sure that the
compounding period and the payment period
match.
 If payments are annual, then you use annual
compounding.
 If payments are monthly, then you use monthly
compounding
6.44
Calculating EARs – Example 1
 To calculate the EAR, use the following formula:
m
 R Nom 
EAR  1 
1

m 

 For example, the EAR for 10% is:
Compounding Periods
EAR
2
10.25%
4
10.3813%
12
10.4713%
52
10.5065%
365
10.5156%
6.45
Decisions, Decisions II
 You are comparing two savings accounts. One pays
5.25%, compounded daily. The other pays 5.3%,
compounded semi-annually. Which account would
you use?
 First account:
 EAR = (1 + .0525/365)365 – 1 = 5.39%
 Second account:
 EAR = (1 + .053/2)2 – 1 = 5.37%
 You should choose the first account (the account that
compounds daily), because you are earning a higher
effective interest rate.
6.46
Decisions, Decisions II Continued
 Let’s verify the choice. Suppose you invest $100 in
each account. How much will you have in each
account in one year?
 First Account:
 Daily rate = .0525 / 365 = .00014383562
 FV = 100(1.00014383562)365 = 105.39, OR,
 365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 105.39
 Second Account:
 Semiannual rate = .053 / 2 = .0265
 FV = 100(1.0265)2 = 105.37, OR,
 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.37
 You have more money in the first account.
6.47
Computing APRs from EARs
 If you have an effective rate, how can you compute
the APR? Rearrange the EAR equation and you get:
1

R Nom  m (1  EAR) m - 1


6.48
APR – Example 1
 Suppose you want to earn an effective rate of 12%
and you are looking at an account that compounds on
a monthly basis. What APR must they pay?
RNom
1

 m (1  EAR) m  1


1

 12 (1  .12) 12  1


 .1138655152 or 11.39%
6.49
Computing Payments with APRs – Example 1
 Suppose you want to buy a new computer system and the store
is willing to allow you to make monthly payments. The entire
computer system costs $3500. The loan period is for 2 years
and the interest rate is 16.9% with monthly compounding.
What is your monthly payment?
 Formula Approach
Calculator Approach
3,500
PV
1  1  r  t 
PVAnnuity  PMT 

0
FV
r


2 x 12 =
N
12 x 2
  .169 

16.9 ÷ 12 = I

1  1 

12


3,500  PMT  
PMT
$172.88
0.169




PMT  $172.88
12


6.50
Future Values with Monthly Compounding
 Suppose you deposit $50 a month into an account that has an
APR of 9%, based on monthly compounding. How much
will you have in the account in 35 years?
Formula Approach
mt


r
 1    1 
m

FVAnnuity  PMT  
r




m


  .09 12 x 35 
 1

 1 
12


 50  
0.09




12


 $147,089.22
Calculator Approach
50
PMT
0
PV
35 x 12 =
N
9 ÷ 12 =
I
FV
$147,089.22
6.51
Mortgages
 In Canada, financial institutions are required by law
to quote mortgage rates with semi-annual
compounding
 Since most people pay their mortgage either monthly
(12 payments per year), semi-monthly (24 payments),
bi-weekly (26 payments), or weekly (52 payments),
you need to remember to convert the interest rate
before calculating the mortgage payment!
 Remember, the payment period and the compounding
period must always be the same.
6.52
The Three Step Process
 Whenever you are faced with a situation when the payment
period and the compounding period are not equal, you can
apply the Three Step Process to solve the problem.
 Step #1: Calculate the Effective Annual Interest Rate, using
the number of compounding periods per year
 Step #2: Calculate a new notional nominal rate (APR), using
the number of payment periods per year
 Step #3: Calculate the per period interest rate by dividing the
answer from Step #2 by the number of payments per year
6.53
Mortgages – Example 1
 Theodore D. Kat is applying to his friendly,
neighbourhood bank for a mortgage of $200,000.
The bank is quoting 6%. He would like to have a 25year amortization period and wants to make
payments monthly. How much are Theodore’s
monthly payments?
6.54
Mortgage – Example 1
 Step #1: Calculate EAR
m
 RNom 
EAR  1 
1

m 

2
 0.06 
 1 
1

2 

 6.09%
 Step #2: Calculate the new notional nominal rate
1

R Nom  m (1  EAR) m - 1


1

 12 (1 .0609) 12 - 1


 5.926346%
6.55
Mortgage – Example 1
 Step #3: Calculate the monthly interest rate by dividing the
answer from Step #2 by the number of payments per year
RMonthly 
RNom
# of payments per year
5.926346%
12
 0.493862% per month

6.56
Mortgage – Example 1
 Complete the problem by solving for the monthly payment,
using the annuity formula
 Formula Approach
1  1  r  t 
PVAnnuity  PMT 

r


1  1.004938622 12 x 25 
200,000  PMT 

.
004938622


PMT  $1,279.61
Calculator Approach
200,000
PV
0
FV
25 x 12 =
N
0.4938622
I
PMT
$1,279.61
6.57
Continuous Compounding
 Sometimes investments or loans are calculated based
on continuous compounding
 EAR = er – 1
 e is the base of the natural logarithms, equal to 2.7183
(although it is actually a non-terminating, non-repeating
decimal)
 e is shown on most calculators ex
 Example: What is the effective annual rate of 7%
compounded continuously?
 EAR = e.07 – 1 = .0725 or 7.25%
6.58
Quick Quiz – Part V
 What is the definition of an APR or nominal
rate?
 What is the effective annual rate?
 Which rate should you use to compare
alternative investments or loans?
6.59
Work the Web Example
 There are web sites available that can easily
provide you with information on home buying
and mortgages
 Click on the web surfer to check out the
Canadian Mortgage and Housing Corporation
site
6.60
Summary 6.5
 You should know how to:
 Calculate PV and FV of multiple cash flows
 Calculate payments
 Calculate PV and FV of regular annuities,
annuities due, growing annuities, perpetuities, and
growing perpetuities
 Calculate EAR’s and effective rates
 Calculate mortgage payments