Download FI3300 Corporation Finance

FI3300
Corporation Finance – Chapter 9
Bond and Stock Valuation
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Learning objectives






Describe the concepts underlying the cost of
capital.
Compute the price of a consol and a preferred
stock.
Compute the price of a zero-coupon bond.
Compute the price of a fixed-coupon bond.
Know the pricing properties of a fixed-coupon
bond.
Compute the price of common stock under
various assumptions about dividend growth.
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Cost of capital 1
 Cost
•
•
•
of capital
How much the firm is willing to pay to get
funds from investors
In other words, it is the cost to the firm for
acquiring money from investors
Usually expressed as a percentage
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Cost of capital 2
•
An investor will provide funds to the firm only if
she earns her required rate of return
•
If investor provides funds to the firm (i.e., a financial
security was bought and sold), the following must be
true:
• Investor earns her required rate of return from the
transaction
• Firm pays just its cost of capital for the funds
• Thus investor’s required rate of return = firm’s
cost of capital
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Cost of capital 3
Capital can be provided either by issuing debt
(e.g., bonds) or equity (common stock).

If debt is issued,
Investor’s required rate of return on debt security
= firm’s cost of debt

If equity is issued,
Investor’s required rate of return on equity security
= firm’s cost of equity
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Blast from the past: last lecture
 Required
rate of return on debt security
is also known as:
•
•
•
Cost of debt
Yield-to-maturity (YTM for short)
Discount rate
 Required
rate of return on equity security
is also known as:
•
•
Cost of equity
Discount rate
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Bond Valuation
Consols, Preferred stock
 Zero-coupon bonds
 Fixed-coupon bonds

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Consols 1



Pays a fixed coupon every period forever.
Has no maturity.
Investor who buys a consol is buying the perpetuity of the
fixed coupon.
So, use PV formula of a perpetuity to find the
present value/price of the consol
Price of consol
=
fixed coupon in dollar ter ms
investor' s required rate of return on consol
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Consols 2

Remember earlier that cost of capital =
investor’s required rate of return. So, we can
re-arrange the equation to find the firm’s cost
of consol capital.
fixed coupon in dollar ter ms

price of consol

Cost of consol capital

We can apply the same ideas to value
preferred stocks. This is because the cash
flows from a preferred stock is also a
perpetuity!
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Preferred stock
Pays a fixed dividend forever.
Price of preferred stock is simply the present
value of a perpetuity.
Preferred stock
Price of
preferred stock
Pps
D

rp
dividend
Required rate of return on preferred stock.
Required rate of
return on preferred
stock/ cost of
capital for preferred
stock
D
rp 
Pps
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Consol problem 1

Problem 9.2
ABC Corp. wants to issue perpetual
debt in order to raise capital. It plans to pay a coupon
of $90 per year on each bond with face value $1,000.
Consols of a comparable firm with a coupon of $100
per year are selling at $1,050. What is the cost of debt
capital for ABC? What will be the price at which it will
issue its consols?

Verify that cost of debt = 0.0952 or 9.52%
Use cost of debt from above to find price of consol.
Verify that price of consol = 90/0.0952 = 945.38

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Consol problem 2

Problem 9.3
If ABC (from the problem above)
wanted to raise $100 million dollars in debt, how many
such consols would it have to issue (to nearest whole
number)?

No. of consols = 100 million/ consol price = 105,778

Problem 9.4
If ABC wanted to issue it’s consols at
par, that is, at a price of $1,000, what coupon must it
pay?
Use coupon = price x required rate of return
Verify that coupon = $95.20 per year


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Zero-coupon bond (ZCB) 1




Call this ZCB for short.
Zero coupon rate, no coupon paid during bond’s life.
Bond holder receives one payment at maturity, the
face value (usually $1000).
Price of a ZCB, PZCB
PZCB 
F = face value of the bond
F
1  r

ZCB N
d
cost of ZCB
debt capital
(in decimals)
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N = number of years
to maturity
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Zero-coupon bond (ZCB) 2
As long as interest rates are positive, the
price of a ZCB must be less than its face
value.
 Why? With positive interest rates, the
present value of the face value (i.e., the
price) has to be less than the face value.

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These problems are just basic TVM problems where you receive
one lump sum in the future.
ZCB Problems
1) Find the price of a ZCB with 20 years to
maturity, par value of $1000 and a required
rate of return of 15% p.a.
N=20, I/Y=15, FV=1000, PMT=0. Price = $61.10
2) XYZ Corp.’s ZCB has a market price of $ 354.
The bond has 16 years to maturity and its face
value is $1000. What is the cost of debt for the
ZCB (i.e., the required rate of return).
PV=-354, FV=1000, N=16, PMT=0.
Required rate of return/ Cost of debt =6.71% p.a.
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Fixed-coupon bond (FCB) 1





Call this FCB for short.
Firm pays a fixed amount (‘coupon’) to the
investor every period until bond matures.
At maturity, firm pays face value of the bond to
investor.
Face value also called par value. Unless
otherwise stated, always assume face value to
be $1000.
Period: can be year, half-year (6 months),
quarter (3 months).
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Fixed-coupon bond (FCB) 2

FCB gives you a stream of fixed payments
plus a lump sum payment (face value) at
maturity.
 This cash flow stream is just an annuity plus a
lump sum at maturity.
 Therefore, we calculate the price of a FCB by
finding the PV of the annuity and lump sum.
 We use the financial calculator to compute the
price of the FCB.
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Fixed-coupon bond (FCB) 2
Price of the FCB, PFCB
Fixed periodic
coupon
Number of
periods to
maturity
Face value
N
PFCB
C
F


t
N
1  rd 
t 1 1  rd 
Cost of debt capital
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Find FCB price

A $1,000 par value bond has coupon
rate of 5% and the coupon is paid semiannually. The bond matures in 20 years
and has a required rate of return of 10%.
Compute the current price of this bond.

PMT = 25. Why?
FV=1000, PMT =25, I/Y=5, N=40. CPT, then PV.
PV = -571.02. Thus, price = $571.02 < par value
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Useful property 1
Go back to the bond in the last problem.
 Suppose annual coupon rate = 10%.
 Verify that price = $1000 = par value

Suppose annual coupon rate = 12%
 Verify that price = $1,171.59 > par value.
It turns out that the following property is true.
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Note: discount rate = cost of debt = required rate of return = yield to maturity
Useful property 2
Coupon rate < discount rate Price < face value Bond is
selling at a
discount
Coupon rate = discount rate Price = face value Bond is
selling at
par
Coupon rate > discount rate Price > face value Bond is
selling at a
premium
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Apply what we learnt

A 10-year annual coupon bond was issued four years
ago at par. Since then the bond’s yield to maturity
(YTM) has decreased from 9% to 7%. Which of the
following statements is true about the current market
price of the bond?




The bond is selling at a discount
The bond is selling at par
The bond is selling at a premium
The bond is selling at book value
Insufficient information

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Try one more

One year ago Pell Inc. sold 20-year, $1,000 par value,
annual coupon bonds at a price of $931.54 per bond.
At that time the market rate (i.e., yield to maturity) was
9 percent. Today the market rate is 9.5 percent;
therefore the bonds are currently selling:

at a discount.
 at a premium.
 at par.
 above the market price.
 not enough information.
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Find YTM, Coupon rate
1)A $1,000 par value bond sells for $863.05. It matures in
20 years, has a 10 percent coupon rate, and pays
interest semi-annually. What is the bond’s yield to
maturity on a per annum basis (to 2 decimal places)?
Verify that YTM = 11.80%
2) ABC Inc. just issued a twenty-year semi-annual coupon
bond at a price of $787.39. The face value of the bond is
$1,000, and the market interest rate is 9%. What is the
annual coupon rate (in percent, to 2 decimal places)?
Verify that annual coupon rate = 6.69%
What happens if bond pays coupon annually? Quarterly?
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Long FCB question


HMV Inc. needs to raise funds for an expansion
project. The company can choose to issue either zerocoupon bonds or semi-annual coupon bonds. In either
case the bonds would have the SAME nominal
required rate of return, a 20-year maturity and a par
value of $1,000. If the company issues the zerocoupon bonds, they would sell for $153.81. If it issues
the semi-annual coupon bonds, they would sell for
$756.32. What annual coupon rate is Camden Inc.
planning to offer on the coupon bonds? State your
answer in percentage terms, rounded to 2 decimal
places.
Verify that annual coupon rate = 7.01%
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Common stock
For common stock, the future cash flows are:
 Dividends
 Selling price
These cash flows are highly uncertain.

To find the value of common stock, we make
assumptions about how dividends evolve in
the future. We look at 3 set of assumptions:
 Constant dividend stream
 Dividends grow at constant rate (constant
dividend growth model)
 Non-constant dividend growth
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Constant dividend stream





Same amount of dividend is paid for ever.
Cash flow stream resembles a perpetuity.
Thus, we value the common stock in the same way as
we value the preferred stock.
Common
Common stock price, Pe
stock
D
Pe 
re
Cost of equity capital, re
D
re 
Pe
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dividend
Cost of equity
capital or
required rate
of return on
equity
27
Dividends grow at constant rate 1
Assume that dividends grow at a
constant rate, g, per period forever.
 Given this assumption, the price of
common stock equals

D0 = Dividend
that the firm just
paid
D0 1  g 
D1
Pe 

re  g
re  g
Required rate
of return on
equity
Don’t panic.
D1 = D0(1 + g)
Dividend
growth rate
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Dividends grow at constant rate 2
Useful properties.
 All other things unchanged,
•
•
•
If D0 increases (decreases), Pe increases
(decreases).
If g increases (decreases), Pe increases
(decreases).
If re increases (decreases), Pe decreases
(increases).
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Dividends grow at constant rate 2
By rearranging the above equation, we
can find the required rate of return on
equity
D1
Capital gains
re 
g
Required rate
yield
of return on
Pe

equity
Dividend yield

For the constant growth model to work,
re > g.
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Constant growth problems 1

Jarrow Company will pay an annual dividend
of $3 per share one year from today. The
dividend is expected to grow at a constant rate
of 7% permanently. The market requires 15%
What is the current price of the stock (to 2
decimal places)?
In this question D1 is already given to you.
Verify that Price = $37.5
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Constant growth problems 2

Johnson Foods Inc. just paid a dividend of $10
(i.e., D0 = 10.00). Its dividends are expected
to grow at a 4% annual rate forever. If you
require a 15% rate of return on investments of
this risk level, what is Johnson Foods’s current
stock price? (to 2 decimal places)
Straightforward application of price formula.
Verify that price = $94.55
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Constant growth problems 3

The price of a stock in the market is $62. You
know that the firm has just paid a dividend of
$5 per share (i.e., D0 = 5). The dividend
growth rate is expected to be 6 percent
forever. What is the investors’ required rate of
return for this stock (to 2 decimal places)?
Use re = (D1/P) + g.
Verify that re = 14.55%
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Constant growth problems 4
A
firm is expected to pay a dividend of
$5.00 on its stock next year. The price of
this stock is $40 and the investor’s
required rate of return is 20%. The firm’s
dividends grow at a constant rate. What
is this constant dividend growth rate (g)?
use re = (D1/P) + g
Verify that g = 7.5%
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Non-constant dividend growth 1
With this assumption, dividends grow at
different rates for different periods of
time. Eventually, dividends will grow at a
constant rate forever.
 Time line is very useful for valuing this
type of stocks.
 To value such stocks, also need the
constant growth formula.
 Best way to learn is through an example.

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Non-constant dividend growth 2

Consider ABC Co.’s dividend stream:
$2.00
T=0

T =1
Dividends grow
at 5% forever
$3.00 $3.50
T=2
T=3
T=4
Discount rate is 15%.
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What to do?
Use constant growth formula to find
stock price at the end of year 3. Call this
stock price P3.
 Add P3 to dividend received at t=3. This
sum is the cash flow for t=3. Find PV of
this cash flow.
 Find PV of dividends at t=1, t=2.
 Current stock price = sum of 2 and 3.

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Apply the method to
find ABC’s stock price
P3 = (3.5 x (1.05))/(0.15 – 0.05) = 36.75
 Find cash flow at t=3

 36.75
+ 3.50 = 40.25
Current stock price, P0
2
3
40.25
P0 


 $30.47
2
3
1.15 1.15
1.15
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Another type of non-constant growth
problem
Malcolm Manufacturing, Inc. just paid a $2.00
annual dividend (that is, D0 = 2.00). Investors
believe that the firm will grow at 10% annually for
the next 2 years and 6% annually forever
thereafter. Assuming a required return of 15%,
what is the current price of the stock (to 2
decimal places)?
Use timeline to ‘see’ the problem better.
Verify that stock price = $25.29
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Summary

Find the price/ present value of debt and
equity securities
 Consols, preferred stock are valued using the
same techniques.
 Fixed-coupon bonds are valued as an annuity
plus a lump sum (face value at maturity)
 Common stocks are valued under 3 different
assumptions about dividends
•
•
•
Constant dividends
Dividends grow at constant rate
Dividends grow at different rates
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