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Chapter 5
Discounted Cash Flow Valuation
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
Chapter Outline
• Future and Present Values of Multiple Cash Flows
• Valuing Level Cash Flows: Annuities and Perpetuities
• Comparing Rates: The Effect of Compounding Periods
• Loan Types and Loan Amortization
1
Basic Rules for Calculating FV and PV of Multiple Cash flows
1. Figure out what the problem is asking. You will get the wrong
answer if you are answering the wrong question.
2. Draw a timeline and label the cash flows and time periods
appropriately.
3. Write out the appropriate formula using symbols first and then
substitute the actual numbers to solve.
4. Check your answers using a calculator.
While these may seem like trivial and time consuming tasks, they will
significantly increase your understanding of the material and your
accuracy rate
2
Future Value of multiple cash flows
Ex: Suppose you invest $7000 in a mutual fund today and $4000 for
each of the next 3 years. If the fund pays 8% annually, how much will
you have in three years? In four years?
0
1
2
$4000
$4000
=
PV (1+r)t
3
4
8%
$7000
Formula: FV
$4000
$0
Find the value at year 3 of each cash flow and add them together.
Today (year 0): FV = $7,000(1.08)3 = $8,817.98
Year 1: FV = $4,000(1.08)2 = $4,665.60
Year 2: FV = $4,000(1.08) = $4,320
Year 3: value = $4,000
Total value in 3 years = $8,817.98 + 4,665.60 + 4,320 + 4,000 =
$21,803.58
Find the value at year 4:
$21,803.58(1.08) = $23,547.87
3
Ex 2: Suppose you invest $500 in a mutual fund today and $600 in one
year.
A. If the fund pays 9% annually, how much will you have in two years?
B. How much will you have in 5 years if you make no further deposits?
0
1
2
3
4
5
$0
$0
$0
$0
9%
$500
$600
A. Formula: FV
=
PV (1+r)t
FV = $500(1.09)2 + $600(1.09) = $1,248.05
B. How much will you have in 5 years if you make no further deposits?
• First way:
FV = $500(1.09)5 + $600(1.09)4 = $1,616.26
• Second way – use value at year 2:
In year 2: FV= $500(1.09)2
$600(1.09)
594.05
+654
$1,248.05
In year 5: FV = $1,248.05(1.09)3 = $1,616.26
4
Ex 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%?
1-7 5-7
Example 3 Time Line
0
1
$100
2
3
4
5
$300
$136.05
$349.92
$485.97
7
In year 5 FV: $100(1.08)4
$136.05
+ $300(1.08)2
+ $349.92
$485.97
5
Present Value of multiple cash flows
Ex: You are considering an investment that will pay you $200,
$400, $600 and $800 in years 1, 2, 3 and 4, respectively. If you
want to earn 12% on your money, how much would you be willing
to pay?
1-9 5-9
Example 5.3 Time Line
0
1
200
2
400
3
600
4
800
178.57
318.88
427.07
508.41
1,432.93
9
• Find the PV of each cash flow and add them
– Year 1 CF: $200 / (1.12)1 = $178.57
– Year 2 CF: $400 / (1.12)2 = $318.88
– Year 3 CF: $600 / (1.12)3 = $427.07
– Year 4 CF: $800 / (1.12)4 = $508.41
– Total PV = $178.57 + 318.88 + 427.07 + 508.41 = $1,432.93
6
Ex: You are considering an investment that will pay you $1,000 in one
year, $2,000 in two years and $3,000 in three years. If you want to earn
10% on your money, how much would you be willing to pay?
 PV = $1,000 / (1.1)1 = $909.09
 PV = $2,000 / (1.1)2 = $1,652.89
 PV = $3,000 / (1.1)3 = $2,253.94
 PV = $909.09 + 1,652.89 + 2,253.94 = $4,815.92
7
Multiple Uneven Cash Flows – Using the Calculator
• Another way to use the financial calculator for uneven cash flows
is to use the cash flow keys
– Texas Instruments BA-II Plus
• Clear the cash flow keys by pressing CF and then 2nd
CLR Work
• Press CF and enter the cash flows beginning with year
0.
• You have to press the “Enter” key for each cash flow
• Use the down arrow key to move to the next cash flow
• The “F” is the number of times a given cash flow
occurs in consecutive years
• Use the NPV key to compute the present value by
[ENTER]ing the interest rate for I, pressing the down
arrow, and then computing NPV
8
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
• CF 2nd ClrWork; CF0 = 0; C01 = 40; F01 = 1; C02 =
75; F02 = 1
• NPV; I = 15; CPT NPV = 91.49
– No! The broker is charging more than you would be willing
to pay.
9
Saving For Retirement
• You are offered the opportunity to put some money away for
retirement. You will receive five annual payments of $25,000 each
beginning in 40 years. How much would you be willing to invest
today if you desire an interest rate of 12%?
– Use cash flow keys:
• CF; CF0 = 0; C01 = 0; F01 = 39; C02 = 25,000; F02 =
5; NPV; I = 12; CPT NPV = 1,084.71
1-15
5-15
Saving For Retirement Time Line
0 1 2
…
39
40
41
42
0 0 0
…
0
25K 25K 25K
43
44
25K 25K
Notice that the year 0 cash flow = 0 (CF0 = 0)
The cash flows years 1 – 39 are 0 (C01 = 0; F01 = 39)
The cash flows years 40 – 44 are 25,000 (C02 = 25,000;
F02 = 5)
15
10
Perpetuities and Annuities
Perpetuity – A stream of level (=) cash payments that never ends
(infinite)
PV of perpetuity = C/r
 Valuing a perpetuity
Where
C = promised cash payment
r = rate of interest on perpetuity
Ex: if you want a perpetuity to pay $100,000 for you and then your
descendents forever. And the interest rate is 10%, what would be the
initial amount paid for this perpetuity?
Note: Initial price  PV of perpetuity
PVperp 
100,000
 $1,000,000
.10
 1 million dollars is the required initial
investment in the perpetuity.
11
A delayed perpetuity – A perpetuity that does not make payments for
several years
Valuing a delayed perpetuity:
Recall: In general: The discount factor
 1 
 (1  r)t 


Accounts for discounting to find the PV of prepayments in the
future
Then
PV of a delayed perpetuity = C x
r
1
(1 +r)t
 In years “t” this becomes an ordinary perpetuity with payments
starting the next year.
Ex: You donate $100,000 to MU to begin 4 years after graduation (with
r = 10%).
1) How much will the endowment be worth in year 3?
2) What is the present value of this endowment?
C 100,000

 $1,000,000
r
.10
endowment = C t  100,000 3  $751,315
r(1  r)
.10(1.10)
1) In year 3 worth =
2) PV of
12
Valuing an Annuity:
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
PV of t-year annuity =
1
1 
C 
t
 r r (1  r ) 
Is this the formula in your book?
1
1 
C 
t
 r r (1  r ) 
Factoring 1/r out of each term
PV of t-year annuity =
We call:
Not quite
1
1 
 r  r (1  r ) t 


1

1  (1  r ) t
C
r







the annuity factor
13
Ex: Annuity pays $4000/year at the end of each of the next 3 years
PV of t-year annuity =
=
1

1

 (1  r ) t
C
r







4000 [(1 – 1/(1.10)3]
r
= 4000 x 2.48685 = $9,947.41
Note: We could have found answer another way
PV 
payment 1 payment 2 payment 3


(1  r)
(1  r) 2
(1  r)3
=
4000
4000
4000


2
(1.10) (1.10)
(1.10)3
= 3,635.36 + 3,305.79 + 3,005.26
= $9,947.41
0
1
10%
0
2
PV
1/ r
3
Time (years)
$4000 $4000 $4000
14
Annuities and Perpetuities – Basic Formulas
• Perpetuity: PV = C / r
• Annuities:
1

1


(1  r ) t
PV  C 
r









 (1  r ) t  1 
FV  C 

r


15