Principles of
Corporate
Finance
Seventh Edition
Chapter 3
How To Calculate Present
Values
Richard A. Brealey
Stewart C. Myers
Slides by
Matthew Will
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 2
Topics Covered
Valuing Long-Lived Assets
PV Calculation Short Cuts
Compound Interest
Nominal and Real Rates of Interest (inflation)
Example: Present Values and Bonds
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 3
Present Values
Discount Factor = DF = PV of $1
DF =
1
t
(1+ r )
Discount Factors can be used to compute
the present value of any cash flow.
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 4
Present Values
C1
PV = DF × C1 =
1 + r1
DF =
1
(1+ r ) t
Discount Factors can be used to compute
the present value of any cash flow.
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 5
Present Values
Ct
PV = DF × C t =
t
(1 + r )
Replacing “1” with “t” allows the formula
to be used for cash flows that exist at any
point in time
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 6
Present Values
Example
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can earn 8% on
your money, how much money should you set aside today
in order to make the payment when due in two years?
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 7
Present Values
Example
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can earn 8% on
your money, how much money should you set aside today
in order to make the payment when due in two years?
PV =
McGraw Hill/Irwin
3000
(1.08 ) 2
= $2,572.02
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 8
Present Values
PVs can be added together to evaluate
multiple cash flows.
PV =
McGraw Hill/Irwin
C1
(1+ r )
C2
1
+ (1+ r ) 2 +....
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 9
Present Values
Given two dollars, one received a year from now
and the other two years from now, the value of each
is commonly called the Discount Factor. Assume r1
= 20% and r2 = 7%.
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 10
Present Values
Given two dollars, one received a year from now
and the other two years from now, the value of each
is commonly called the Discount Factor. Assume r1
= 20% and r2 = 7%.
McGraw Hill/Irwin
DF1 =
1.00
(1+.20 )1
= .83
DF2 =
1.00
(1+.07 ) 2
= .87
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 11
Present Values
Example
Assume that the cash flows
from the construction and sale
of an office building is as
follows. Given a 7% required
rate of return, create a present
value worksheet and show the
net present value.
Year 0
Year 1
Year 2
− 150,000 − 100,000 + 300,000
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 12
Present Values
Example - continued
Assume that the cash flows from the construction and sale of an office
building is as follows. Given a 7% required rate of return, create a
present value worksheet and show the net present value.
Period
0
1
2
McGraw Hill/Irwin
Discount
Factor
1.0
1
1.07 = .935
1
(1.07 )2
= .873
Cash
Flow
− 150,000
− 100,000
Present
Value
− 150,000
− 93,500
+ 300,000
+ 261,900
NPV = Total =
$18,400
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 13
Short Cuts
Sometimes there are shortcuts that make it
very easy to calculate the present value of an
asset that pays off in different periods. These
tolls allow us to cut through the calculations
quickly.
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 14
Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cash flow
Return =
present value
C
r=
PV
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 15
Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cash flow
PV of Cash Flow =
discount rate
C1
PV =
r
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 16
Short Cuts
Perpetuity – Zahlungen steigen jährlich mit
konstanter Wachstumsrate g.
cash flow
PV Cash Flow =
discount rate - growth rate
C1
PV =
r−g
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 17
Short Cuts
Annuity - An asset that pays a fixed sum each
year for a specified number of years.
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 18
Short Cuts
Annuity - An asset that pays a fixed sum each
year for a specified number of years.
1
1
PV of annuity = C × −
t
r r (1 + r )
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 19
Annuity Short Cut
Example
You agree to lease a car for 4 years at $300 per month.
You are not required to pay any money up front or at the
end of your agreement. If your opportunity cost of capital
is 0.5% per month, what is the cost of the lease?
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 20
Annuity Short Cut
Example - continued
You agree to lease a car for 4 years at $300 per
month. You are not required to pay any money up
front or at the end of your agreement. If your
opportunity cost of capital is 0.5% per month,
what is the cost of the lease?
1
1
Lease Cost = 300 ×
−
48
.005 .005(1 + .005)
Cost = $12,774.10
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 21
Compound Interest
i
ii
Periods Interest
per
per
year
period
iii
APR
(i x ii)
iv
Value
after
one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
McGraw Hill/Irwin
6.000%
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 22
McGraw Hill/Irwin
Compound Interest
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 23
18
16
14
12
10
8
6
4
2
0
10% Simple
30
27
24
21
18
15
12
9
6
10% Compound
3
0
FV of $1
Compound Interest
Number of Years
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 24
Compound Interest
Example
Suppose you are offered an automobile loan at an APR of
6% per year. What does that mean, and what is the true
rate of interest, given monthly payments?
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 25
Compound Interest
Example - continued
Suppose you are offered an
automobile loan at an APR of 6% per
year. What does that mean, and what
is the true rate of interest, given
monthly payments? Assume $10,000
loan amount.
Loan Pmt = 10,000 × (1.005)
= 10,616.78
APR = 6.1678%
McGraw Hill/Irwin
12
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 26
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.
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 27
Inflation
1+ nominal interest rate
1 + real interest rate =
1+inflation rate
approximation formula
Real int. rate ≈ nominal int. rate - inflation rate
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 28
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
1+.059
1 + real interest rate = 1+.033 Savings
1 + real interest rate =
real interest rate
=
1.025
Bond
.025 or 2.5%
Approximation =.059-.033 =.026 or 2.6%
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 29
Valuing a Bond
Example
If today is October 2002, what is the value of the following
bond?
An IBM Bond pays $115 every Sept for 5 years. In Sept
2007 it pays an additional $1000 and retires the bond.
The bond is rated AAA (WSJ AAA YTM is 7.5%)
Cash Flows
Sept 03 04 05 06 07
115 115 115 115 1115
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 30
Valuing a Bond
Example continued
If today is October 2002, what is the value of the following bond?
An IBM Bond pays $115 every Sept for 5 years. In Sept 2007 it pays an
additional $1000 and retires the bond.
The bond is rated AAA (WSJ AAA YTM is 7.5%)
115
115
115
115
1,115
PV =
+
+
+
+
2
3
4
1.075 (1.075) (1.075) (1.075) (1.075)5
= $1,161.84
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 31
Bond Prices and Yields
1600
1400
Price
1200
1000
800
600
400
200
0
0
2
4
6
5 Year 9% Bond
McGraw Hill/Irwin
8
10
12
14
Yield
1 Year 9% Bond
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 32
Aufgaben in der Vorlesung
Q2: BW von $139 beträgt $125. Abzinsungsfaktor?
Q4: BW von $374, die in Jahr 9 gezahlt werden?
Kapitalkostensatz 9%.
Q5: Projekt bringt in den Jahren 1, 2 und 3 $432, $137
und $797. BW bei Kapitalkosten von 15%?
Q7: NBW eines Projekts mit Anfangsauszahlung $1548
und unendlichen jährlichen EZÜ von $138. Zinssatz: 9%.
Q8: Aktie zahlt nächstes Jahr $4 Dividende, danach jedes
Jahr 4% mehr. BW Dividenden bei Zinssatz 14%?
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 33
Q12: Nominaler Zinssatz 25%, Inflationsrate 21%.
Realer Zinssatz?
Q14: Anlage von $10 Mio. zu 6%. Wert nach vier Jahren
bei
a) jährlicher Verzinsung?
b) monatlicher Verzinsung?
c) kontinuierlicher Verzinsung?
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
3- 34
Aufgaben zu Hause
PQ4, 7, 8, 21, 31, 33
McGraw Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved
