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8/2/2019 Ross Chap004 Rev
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Discounted Cash Flow Valuation
Chapter 4
Copyright 2010 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
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Key Concepts and Skills Be able to compute the future value and/or
present value of a single cash flow or series of
cash flows Be able to compute the return on an
investment
Be able to use a financial calculator and/orspreadsheet to solve time value problems
Understand perpetuities and annuities
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Chapter Outline4.1 Valuation: The One-Period Case
4.2 The Multiperiod Case
4.3 Compounding Periods
4.4 Simplifications
4.5 Loan Amortization
4.6 What Is a Firm Worth?
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4.1 The One-Period Case If you were to invest $10,000 at 5-percent interest
for one year, your investment would grow to$10,500.
$500 would be interest ($10,000 .05)
$10,000 is the principal repayment ($10,000 1)
$10,500 is the total due. It can be calculated as:
$10,500 = $10,000(1.05)
The total amount due at the end of the investment is
call theFuture Value (FV).
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Future Value In the one-period case, the formula forFVcan
be written as:
FV= C0(1 + r)
Where C0 is cash flow today (time zero), and
ris the appropriate interest rate.
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Present Value If you were to be promised $10,000 due in one year
when interest rates are 5-percent, your investment
would be worth $9,523.81 in todays dollars.
05.1
000,10$81.523,9$ !
The amount that a borrower would need to set aside
today to be able to meet the promised payment of$10,000 in one year is called the Present Value (PV).
Note that $10,000 = $9,523.81(1.05).
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Present Value
In the one-period case, the formula forPVcanbe written as:
r
CPV
!1
1
Where C1 is cash flow at date 1, and
ris the appropriate interest rate.
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Net Present Value The Net Present Value (NPV) of an
investment is the present value of the
expected cash flows, less the cost of theinvestment.
Suppose an investment that promises to pay
$10,000 in one year is offered for sale for$9,500. Your interest rate is 5%. Should you
buy?
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Net Present Value
81.23$
81.523,9$500,9$
05.1
000,10$500,9$
!!
!
NPV
NPV
NPV
The present value of the cash inflow is greaterthan the cost. In other words, the Net Present
Value is positive, so the investment should be
purchased.
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Net Present ValueIn the one-period case, the formula forNPVcan be
written as:
NPV= Cost+ PV
If we had notundertaken the positive NPVproject
considered on the last slide, and instead invested our
$9,500 elsewhere at 5 percent, ourFVwould be less
than the $10,000 the investment promised, and we
would be worse off inFVterms :
$9,500(1.05) = $9,975 < $10,000
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4.2 The Multiperiod Case The general formula for the future value of an
investment over many periods can be written
as:FV= C0(1 + r)
T
Where
C0 is cash flow at date 0,
ris the appropriate interest rate, and
Tis the number of periods over which the cash isinvested.
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Future Value
Suppose a stock currently pays a dividend of
$1.10, which is expected to grow at 40% per
year for the next five years.
What will the dividend be in five years?
FV= C0(1 + r)T
$5.92 = $1.10(1.40)5
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Future Value and Compounding
Notice that the dividend in year five, $5.92,
is considerably higher than the sum of the
original dividend plus five increases of 40-
percent on the original $1.10 dividend:
$5.92 > $1.10 +5[$1.10.40] = $3.30
This is due to compounding.
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Future Value and Compounding
0 1 2 3 4 5
10.1$
3)40.1(10.1$ v
02.3$
)40.1(10.1$ v
54.1$
2)40.1(10.1$ v
16.2$
5)40.1(10.1$ v
92.5$
4)40.1(10.1$ v
23.4$
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Present Value and Discounting How much would an investor have to set
aside today in order to have $20,000 five
years from now if the current rate is 15%?
0 1 2 3 4 5
$20,000PV
5)15.1(
000,20$53.943,9$ !
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4.5 Finding the Number of PeriodsIf we deposit $5,000 today in an account paying 10%,
how long does it take to grow to $10,000?T
rCFV )1(0 v!T
)10.1(000,5$000,10$ v!
2000,5$
000,10$)10.1( !!T
)2ln()10.1ln( !T
years27.7
0953.0
6931.0
)10.1ln(
)2ln(!!!T
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Assume the total cost of a college education will be$50,000 when your child enters college in 12 years.You have $5,000 to invest today. What rate of interest
must you earn on your investment to cover the cost ofyour childs education?
What Rate Is Enough?
TrCFV )1(0 v!
12)1(000,5$000,50$ rv!
10000,5$
000,50$)1( 12 !! r 12110)1( ! r
2115.12115.1110121
!!!r
About 21.15%.
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Calculator Keys Texas Instruments BA-II Plus
FV = future value
PV = present value I/Y = periodic interest rate
P/Y must equal 1 for the I/Y to be the periodic rate
Interest is entered as a percent, not a decimal
N = number of periods Remember to clear the registers (CLR TVM) after
each problem
Other calculators are similar in format
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Multiple Cash Flows Consider an investment that pays $200 one
year from now, with cash flows increasing by
$200 per year through year 4. If the interestrate is 12%, what is the present value of this
stream of cash flows?
If the issuer offers this investment for $1,500,should you purchase it?
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Multiple Cash Flows0 1 2 3 4
200 400 600 800178.57
318.88
427.07
508.41
1,432.93
Present Value < Cost Do Not Purchase
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Valuing Lumpy Cash FlowsFirst, set your calculator to 1 payment per year.
Then, use the cash flow menu:
CF2
CF1
F2
F1
CF0
1
200
1
1,432.93
0
400
I
NPV
12
CF4
CF3
F4
F3 1
600
1
800
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4.3 Compounding PeriodsCompounding an investment m times a year for
Tyears provides for future value of wealth:
Tm
m
rCFV
v
v! 10
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Compounding Periods For example, if you invest $50 for 3 years at
12% compounded semi-annually, your
investment will grow to
93.70$)06.1(50$2
12.150$ 6
32
!v!
v!
v
FV
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Effective Annual Rates of InterestA reasonable question to ask in the above
example is what is the effective annualrate of
interest on that investment?
The Effective Annual Rate (EAR) of interest isthe annual rate that would give us the same
end-of-investment wealth after 3 years:
93.70$)06.1(50$)2
12.1(50$ 632 !v!v! vFV
93.70$)1(50$
3 !v EAR
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Effective Annual Rates of Interest
So, investing at 12.36% compounded annually
is the same as investing at 12% compounded
semi-annually.
93.70$)1(50$ 3 !v! EARFV
50$
93.70$)1( 3 ! EAR
1236.150$
93.70$31
!
!EAR
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Effective Annual Rates of Interest Find the Effective Annual Rate (EAR) of an
18% APR loan that is compounded monthly.
What we have is a loan with a monthly
interest rate rate of 1%.
This is equivalent to a loan with an annual
interest rate of 19.56%.
1956.1)015.1(12
18.11 12
12
!!
!
m
m
r
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EAR on a Financial Calculator
keys: description:
[2nd] [ICONV] Opens interest rate conversion menu
[] [EFF=] [CPT] 19.56
Texas Instruments BAII Plus
[][NOM=] 18 [ENTER] Sets 18 APR.[] [C/Y=] 12 [ENTER] Sets 12 payments per year
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Continuous Compounding The general formula for the future value of an
investment compounded continuously over manyperiods can be written as:
FV= C0erT
Where
C0 is cash flow at date 0,
ris the stated annual interest rate,
Tis the number of years, and
e is a transcendental number approximately equal
to 2.718. ex is a key on your calculator.
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4.4 Simplifications Perpetuity
A constant stream of cash flows that lasts forever
Growing perpetuity A stream of cash flows that grows at a constant rate
forever
Annuity
A stream of constant cash flows that lasts for a fixednumber of periods
Growing annuity
A stream of cash flows that grows at a constant rate for
a fixed number of periods
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PerpetuityA constant stream of cash flows that lasts forever
0
1
C
2
C
3
C
.! 32 )1()1()1( r
C
r
C
r
C
PV
r
CPV !
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Perpetuity: ExampleWhat is the value of a British consol that
promises to pay 15 every year for ever?
The interest rate is 10-percent.
0
1
15
2
15
3
15
15010.
15!!PV
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Growing PerpetuityA growing stream of cash flows that lasts forever
0
1
C
2
C(1+g)
3
C(1+g)2
.
v
v
! 3
2
2 )1(
)1(
)1(
)1(
)1( r
gC
r
gC
r
C
PV
gr
CPV
!
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Growing Perpetuity: ExampleThe expected dividend next year is $1.30, and
dividends are expected to grow at 5% forever.
If the discount rate is 10%, what is the value of thispromised dividend stream?
0
1
$1.30
2
$1.30(1.05)
3
$1.30(1.05)2
00.26$05.10.
30.1$!
!PV
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AnnuityA constant stream of cash flows with a fixed maturity
0 1
C
2
C
3
C
Tr
C
r
C
r
C
r
C
PV )1()1()1()1( 32 ! .
-
!T
rr
CPV
)1(
11
T
C
.
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Annuity: ExampleIf you can afford a $400 monthly car payment, howmuch car can you afford if interest rates are 7% on 36-month loans?
0 1
$400
2
$400
3
$400
59.954,12$
)1207.1(
11
12/07.
400$36
!
-
!PV
36
$400
.
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What is the present value of a four-year annuity of $100 per
year that makes its first payment two years from today if the
discount rate is 9%?
22.297$09.1
97.327$
0!!
PV
0 1 2 3 4 5
$100 $100 $100 $100$323.97$297.22
97.323$)09.1(
100$
)09.1(
100$
)09.1(
100$
)09.1(
100$
)09.1(
100$4321
4
1
1 !!!!t
tPV
2-35
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Growing AnnuityA growing stream of cash flows with a fixed maturity
0 1
C
T
T
r
gC
r
gC
r
CPV )1(
)1(
)1(
)1(
)1(
1
2
v
v
!
.
-
!
T
r
g
gr
CPV
)1(
11
.
2
C(1+g)
3
C(1+g)2
T
C(1+g)T-1
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Growing Annuity: ExampleA defined-benefit retirement plan offers to pay $20,000 peryear for 40 years and increase the annual payment by 3% eachyear. What is the present value at retirement if the discount rateis 10%?
0 1
$20,000
57.121,265$
10.1
03.11
03.10.
000,20$40
!
-
!PV
.
2
$20,000(1.03)
40
$20,000(1.03)39
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Growing Annuity: ExampleYou are evaluating an income generating property. Net rent is
received at the end of each year. The first year's rent is
expected to be $8,500, and rent is expected to increase 7%
each year. What is the present value of the estimated incomestream over the first 5 years if the discount rate is 12%?
0 1 2 3 4 5
500,8$
!v )07.1(500,8$!v 2)07.1(500,8$
095,9$65.731,9$
!v 3)07.1(500,8$
87.412,10$
!v 4)07.1(500,8$
77.141,11$
$34,706.26
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4.5 Loan Amortization Pure Discount Loans are the simplest form of loan.
The borrower receives money today and repays a
single lump sum (principal and interest) at a future
time.
Interest-Only Loans require an interest payment each
period, with full principal due at maturity.
Amortized Loans require repayment of principalover time, in addition to required interest.
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Pure Discount Loans Treasury bills are excellent examples of pure
discount loans. The principal amount is repaid
at some future date, without any periodicinterest payments.
If a T-bill promises to repay $10,000 in 12
months and the market interest rate is 7percent, how much will the bill sell for in the
market?
PV = 10,000 / 1.07 = 9,345.79
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Interest-Only Loan Consider a 5-year, interest-only loan with a 7%
interest rate. The principal amount is $10,000.
Interest is paid annually. What would the stream of cash flows be?
Years 1 4: Interest payments of .07(10,000) = 700
Year 5: Interest + principal = 10,700
This cash flow stream is similar to the cash
flows on corporate bonds, and we will talk
about them in greater detail later.
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Amortized Loan with Fixed Principal
Payment Consider a $50,000, 10 year loan at 8%
interest. The loan agreement requires the firm
to pay $5,000 in principal each year plusinterest for that year.
Click on the Excel icon to see the amortization
table
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Amortized Loan with Fixed Payment Each payment covers the interest expense plus reduces
principal
Consider a 4 year loan with annual payments. Theinterest rate is 8% ,and the principal amount is $5,000.
What is the annual payment?
4 N
8 I/Y
5,000 PV
CPT PMT = -1,509.60
Click on the Excel icon to see the amortization table
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4.6 What Is a Firm Worth? Conceptually, a firm should be worth the
present value of the firms cash flows.
The tricky part is determining the size, timing,and riskof those cash flows.
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Quick Quiz How is the future value of a single cash flow
computed?
How is the present value of a series of cash flowscomputed.
What is the Net Present Value of an investment?
What is an EAR, and how is it computed?
What is a perpetuity? An annuity?