Cash Chapter4
Discounted Flow Valuation
McGraw-Hill/Irwin
Copyright 2010 by the McGraw-Hill Companies, Inc. All rights reserved.
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/or
spreadsheet to solve time value problems
Understand perpetuities and annuities
4-2
Chapter Outline
4.1 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?
4-3
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 the Future Value (FV).
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Future Value
In the one-period case, the formula for FV can
be written as:
FV = C0(1 + r)
Where C0 is cash flow today (time zero), and
r is the appropriate interest rate.
4-5
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.
$10,000
$9,523.81 =
1.05
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 for PV can
be written as:
C1
PV =
1+ r
Where C1 is cash flow at date 1, and
r is the appropriate interest rate.
4-7
Net Present Value
The Net Present Value (NPV) of an
investment is the present value of the
expected cash flows, less the cost of the
investment.
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?
4-8
Net Present Value
$10,000
NPV = $9,500 +
1.05
NPV = $9,500 + $9,523.81
NPV = $23.81
The present value of the cash inflow is greater
than the cost. In other words, the Net Present
Value is positive, so the investment should be
purchased.
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Net Present Value
In the one-period case, the formula for NPV can be
written as:
NPV = Cost + PV
If we had not undertaken the positive NPV project
considered on the last slide, and instead invested our
$9,500 elsewhere at 5 percent, our FV would be less
than the $10,000 the investment promised, and we
would be worse off in FV terms :
$9,500(1.05) = $9,975 < $10,000
4-10
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,
r is the appropriate interest rate, and
T is the number of periods over which the cash is
invested.
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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
4-12
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 40percent on the original $1.10 dividend:
$5.92 > $1.10 + 5[$1.10.40] = $3.30
This is due to compounding.
4-13
Future Value and Compounding
$1.10 (1.40)
5
$1.10 (1.40) 4
$1.10 (1.40) 3
$1.10 (1.40) 2
$1.10 (1.40)
$1.10
0
$1.54 $2.16
1
2
$3.02
$4.23
$5.92
3
4
5
4-14
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%?
PV
$20,000
0
1
2
3
4
5
$20,000
$9,943.53 =
5
(1.15)
4-15
4.5 Finding the Number of Periods
If we deposit $5,000 today in an account paying 10%,
how long does it take to grow to $10,000?
FV = C0 (1 + r )
$10,000 = $5,000 (1.10)
T
T
$10,000
(1.10) =
=2
$5,000
T
ln(1.10)T = ln(2)
ln(2)
0.6931
T=
=
= 7.27 years
ln(1.10) 0.0953
4-16
What Rate Is Enough?
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 of
your childs education?
About 21.15%.
FV = C0 (1 + r )
T
$50,000
(1 + r ) =
= 10
$5,000
12
r = 10
1 12
$50,000 = $5,000 (1 + r )12
(1 + r ) = 101 12
1 = 1.2115 1 = .2115
4-17
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 interest
rate 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?
4-18
Multiple Cash Flows
0
1
200
2
3
4
400
600
800
178.57
318.88
427.07
508.41
1,432.93
Present Value < Cost Do Not Purchase
4-19
4.3 Compounding Periods
Compounding an investment m times a year for
T years provides for future value of wealth:
r
FV = C0 1 +
m
mT
4-20
Compounding Periods
For example, if you invest $50 for 3 years at
12% compounded semi-annually, your
investment will grow to
.12
FV = $50 1 +
2
23
= $50 (1.06) 6 = $70.93
4-21
Effective Annual Rates of Interest
A reasonable question to ask in the above
example is what is the effective annual rate of
interest on that investment?
.12 23
FV = $50 (1 +
) = $50 (1.06) 6 = $70.93
2
The Effective Annual Rate (EAR) of interest is
the annual rate that would give us the same
end-of-investment wealth after 3 years:
$50 (1 + EAR) 3 = $70.93
4-22
Effective Annual Rates of Interest
FV = $50 (1 + EAR) 3 = $70.93
$70.93
(1 + EAR) =
$50
3
13
$70.93
EAR =
1 = .1236
$50
So, investing at 12.36% compounded annually
is the same as investing at 12% compounded
semi-annually.
4-23
Effective Rates Annual 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%.
m
12
r
.18
1 + = 1 +
= (1.015)12 = 1.1956
m
12
4-24
Continuous Compounding
The general formula for the future value of an
investment compounded continuously over many
periods can be written as:
FV = C0erT
Where
C0 is cash flow at date 0,
r is the stated annual interest rate,
T is the number of years, and
e is a transcendental number approximately equal
to 2.718. ex is a key on your calculator.
4-25
4.4 Simplifications
Perpetuity
Growing perpetuity
A stream of cash flows that grows at a constant rate
forever
Annuity
A constant stream of cash flows that lasts forever
A stream of constant cash flows that lasts for a fixed
number of periods
Growing annuity
A stream of cash flows that grows at a constant rate for
a fixed number of periods
4-26
Perpetuity
A constant stream of cash flows that lasts forever
C
0
C
C
1
2
3
C
C
C
PV =
+
+
+
2
3
(1 + r ) (1 + r ) (1 + r )
C
PV =
r
4-27
Perpetuity: Example
What is the value of a British consol that
promises to pay 15 every year for ever?
The interest rate is 10-percent.
15
0
15
15
1
2
3
15
PV =
= 150
.10
4-28
Growing Perpetuity
A growing stream of cash flows that lasts forever
C
0
C(1+g)
C (1+g)2
1
2
3
C
C (1 + g ) C (1 + g )
PV =
+
+
+
2
3
(1 + r )
(1 + r )
(1 + r )
2
C
PV =
rg
4-29
Growing Perpetuity: Example
The 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 this
promised dividend stream?
$1.30
$1.30(1.05)
$1.30 (1.05)2
0
1
2
3
$1.30
PV =
= $26.00
.10 .05
4-30
Annuity
A constant stream of cash flows with a fixed maturity
C
C
C
C
0
1
2
3
T
C
C
C
C
PV =
+
+
+
2
3
T
(1 + r ) (1 + r ) (1 + r )
(1 + r )
C
1
PV = 1
T
r (1 + r )
4-31
Annuity: Example
If you can afford a $400 monthly car payment, how
much car can you afford if interest rates are 7% on 36month loans?
$400
$400
$400
$400
0
1
2
3
36
$400
1
PV =
1 (1 + .07 12) 36 = $12,954.59
.07 / 12
4-32
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%?
4
PV1 =
t =1
$297.22
0
$100
$100
$100
$100
$100
=
+
+
+
= $323.97
t
1
2
3
4
(1.09) (1.09) (1.09) (1.09) (1.09)
$323.97
1
$100
2
$327.97
PV =
= $297.22
0
1.09
$100
3
$100
4
$100
5
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Growing Annuity
A growing stream of cash flows with a fixed maturity
C
C(1+g)
C (1+g)2
C(1+g)T-1
0
1
2
3
T
C
C (1 + g )
C (1 + g )T 1
PV =
+
++
2
T
(1 + r )
(1 + r )
(1 + r )
C
PV =
rg
1+ g
1
(1 + r )
T
4-34
Growing Annuity: Example
A defined-benefit retirement plan offers to pay $20,000 per
year for 40 years and increase the annual payment by 3% each
year. What is the present value at retirement if the discount rate
is 10%?
$20,000
$20,000(1.03) $20,000(1.03)39
0
1
2
40
1.03 40
$20,000
PV =
= $265,121.57
1
.10 .03 1.10
4-35
Growing Annuity: Example
You 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 income
stream over the first 5 years if the discount rate is 12%?
$8,500 (1.07) 2 =
$8,500 (1.07) 4 =
$8,500 (1.07) =
$8,500 (1.07) 3 =
$8,500 $9,095 $9,731.65 $10,412.87 $11,141.77
0
1
$34,706.26
2
3
4
5
4-36
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 principal
over time, in addition to required interest.
4-37
Pure Discount Loans
Treasury bills are excellent examples of pure
discount loans. The principal amount is repaid
at some future date, without any periodic
interest payments.
If a T-bill promises to repay $10,000 in 12
months and the market interest rate is 7
percent, how much will the bill sell for in the
market?
PV = 10,000 / 1.07 = 9,345.79
4-38
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.
4-39
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 plus
interest for that year.
Click on the Excel icon to see the amortization
table
4-40
Amortized Loan with Fixed Payment
Each
payment covers the interest expense plus reduces
principal
Consider a 4 year loan with annual payments. The
interest rate is 8% ,and the principal amount is $5,000.
What is the annual payment?
Click
4N
8 I/Y
5,000 PV
CPT PMT = -1,509.60
on the Excel icon to see the amortization table
4-41
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 risk of those cash flows.
4-42
Quick Quiz
How is the future value of a single cash flow
computed?
How is the present value of a series of cash flows
computed.
What is the Net Present Value of an investment?
What is an EAR, and how is it computed?
What is a perpetuity? An annuity?
4-43