BFF2140 Corporate Finance 1
Lecture 3: Valuation of bonds and equities
Chief Examiner/Lecturer: Dr Amale Scally
MONASH
BUSINESS
SCHOOL
Teaching Week Three
Valuation of Bonds and Equities
Readings
Chapter 6, pp. 153-171
Chapter 7, pp. 189-205
MONASH
BUSINESS
SCHOOL
Learning Objectives
•
Understand and implement bond valuation techniques.
•
Develop an understanding of bond concepts and how interest rate
risk impacts on bond pricing.
•
Estimate the value of shares with zero, constant and varying
growth.
•
Comprehend the issues associated with the models employed in
share valuation.
Bond Valuation
What is a bond?
Source
What is a bond?
▪ Bonds are included in the “right hand side” of the balance sheet
A = L+ E
▪ They specifically form a part of debt (liabilities).
▪ Large companies commonly issue bonds to raise funds in the
market.
▪ Investors therefore lend their money to the company and receive
interest payments (called coupons) until such a point in future
where the bond matures and the principal is repaid.
What is a bond?
So there are TWO cash flow streams associated with bonds:
1. Repayment of principal at maturity (value stated on bond)
2. Coupon interest payments throughout the life of the bonds.
As with anything else the price of the bond today is simply the PV
of ALL future cash flows.
Valuation of Bonds and Equities
• Intrinsic Value
The value of financial securities = PV of all expected future cash flows.
• Thus, to value bonds or stocks, we need to
– Estimate future cash flows:
• Amount (how much) and
• Timing (when)
– Then discount future cash flows at an appropriate rate:
• The rate should be appropriate to the risk associated with the security.
Definition and Features of a Bond
A bond is a certificate showing that a borrower owes a specified
sum (the principal / face value) and interest payments (in the case
of coupon paying bonds).
Features and notation:
– Coupon payments (CPN): The stated interest payments. Payment is
constant and usually payable every year or half year.
– Coupon rate: The annual coupon divided by the face value.
– Face value or par value (FV): The principal amount repayable at the
end of the term.
– Maturity (n): The specified date at which the principal amount is
payable.
– Required Return on Bond (y): the return demanded by a bondholder
for investing in a debt security given its level of risk.
Definition and Features of a Bond
• Bond issuer
– Person who issues the bond and must repay the face value
at maturity, i.e. the borrower
• Bond holder
– Person who holds the bond certificate and will receive the
face value at maturity
• A bond is a tradeable financial instrument
– i.e. a bond can be bought and sold
Cash Flows for a Bond
• Notation:
Coupon (CPN or C)
Face Value (FV or F)
Time to Maturity (n)
• Cash flows of a typical bond:
P
CPN
CPN
CPN
CPN
0
1
2
3
n-1
CPN+FV
n=maturity
Important!!
A bond will therefore trade many times in its life
AND
The bond price will change very regularly …
Bond Value
Bond Value (P0) = PV(Coupons) + PV(Face Value)
CPN 
1
P0 =
1 −
y  (1 + y )n

FV
+
n
 (1 + y )
Note y here is yield – similar to i. It is really a proxy for the
required rate of return to debt holders, the cost of debt.
Pure Discount (zero coupon) Bonds
Information needed for valuing pure discount bonds:
–
–
–
–
Time to maturity (n) = Maturity date - today’s date
Compounding frequency (m)
Face value (FV)
Discount rate (y)
$0
$0
$0
$ FV

0
n −1
2
1
Present value of a pure discount bond at time 0:
FVmn
P =
y mn
(1 +
)
m
n
Example 1: Pure Discount Bonds
Find the value of a 30-year zero-coupon bond with a $1,000 par
value and a required rate of return of 6%.
$0
$0
$0
$ 1,000
$0$1,0

102 3029

0
1
2
29
FV
$1,000
P =
=
= $174.11
n
30
(1 + y)
(1.06)
30
Example 1: Pure Discount Bonds
– Continued (using HP10bII+ calculator )
Coupon Bonds
Information needed to value coupon bonds:
– Coupon payment dates and time to maturity (n)
– Coupon payment (CPN) per period and face value (FV)
– Discount rate (y)
$CPN
$CPN
$CPN
$CPN + $FV
n −1
n

0
1
2
Coupon Bonds
Value of a level coupon bond
= PV of coupon payment annuity + PV of face value
CPN 
1 
FV
P =
+
1 −
n 
n
y  (1 + y )  (1 + y )
Example 2: Bond Valuation (with coupon rate = y)
A bond with a face value of $1,000 and a coupon rate of
5% has 10 years to maturity. What is the market price of
this bond if the discount rate is 5%?
 $1,000
$50 
1
P =
1−
+

10 
0.05  (1.05)  (1.05)10
P = $1,000.00
• If the coupon rate is the same as the discount rate (y), then the
bond trades AT PAR.
• Using Financial tables:
P = $50 x PVIFA105 + $1,000 x PVIF105
= $50 x 7.722 + $1,000 x 0.614 = $1,000.10 (rounding error)
Example 2: Coupon Bonds
– Continued (using HP10bII+ calculator )
Example 3: Bond Valuation (with coupon rate < y)
A bond with a face value of $1,000 and a coupon rate of
5% has 10 years to maturity. What is the market price of
this bond if the discount rate is 6%?
$50 
1  $1,000
P =
1−
+

10 
0.06  (1.06)  (1.06)10
P = $926.40
• If the coupon rate is less than the discount rate (y), then the bond
trades at A DISCOUNT.
• Using Financial tables:
P = $50 x PVIFA106 + $1,000 x PVIF106
= $50 x 7.360 + $1,000 x 0.558 = $926.00 (rounding error)
Example 3: Coupon Bonds
– Continued (using HP10IIb+ calculator )
Example 4: Bond Valuation (with coupon rate > y)
A bond with a face value of $1,000 and a coupon rate of
5% has 10 years to maturity. What is the market price of
this bond if the discount rate is 4%?
 $1,000
$50 
1
P=
1−
+

10 
0.04  (1.04)  (1.04)10
P = $1,081.11
• If the coupon rate is greater than the discount rate (y), then
the bond trades at A PREMIUM.
• Using Financial tables:
P = $50 x PVIFA104 + 1000 x PVIF104
= $50×8.111+$1000×0.676 = $1081.55 (rounding error)
Example 4: Coupon Bonds
– Continued (using HP10IIb+ calculator )
Example 5: Semi-annual Coupon Bonds
A bond with a face value of $1,000 and a coupon rate of 6% has
10 years to maturity. What is the market price of this bond if the
discount rate is 10%? Assume coupon payments are paid semiannually (i.e. half yearly)
Define period=6 months and convert CPN, y and n accordingly
 $1,000
$30 
1
B=
1−
+

20 
0.05  (1.05)  (1.05) 20
B = $750.76
Using Financial tables:
B0 = $30 x PVIFA205 + $1,000 x PVIF205
= $30×12.462 + 1000×0.377 = $750.86 (rounding error)
Example 5: Semi Annual Coupon Bonds
– Continued (using HP10bII+ calculator )
The Effective Annual Yield
Interest rates should always be quoted on an annualised
basis.
Recall that the effective annual interest rate (EAR)
measures the return of $1 in one year.
What is the EAR (also referred to as EAY-effective annual yield)
for the bond in example 5? Recall a bond with a face value of
$1,000; a coupon rate of 6%; 10 years to maturity; and a yield to
maturity of 10% with semi-annual coupon payments.
EAY = (1+ 0.10/2)2 - 1 = 0.1025 or 10.25% p.a.
Bond Yields
Yield to maturity is the interest rate that equates a bond’s present
value of interest payments and principal repayment with its price.
– This can be viewed as follows:
“The YTM measures the average rate of return obtained by
investors if the bond is purchased now and held until
maturity and if there is no default on any of the promised
payments.”
• There is an inverse relationship between market interest rates and
bond price.
Why? Because the bond price is a present value and there is
always a negative relationship between PV and interest rates.
YTM and Bond Value
Assume face value = $1000 and N = 4 years
$1400
When the YTM < coupon, the bond trades
at a premium.
Bond Value
1300
When the YTM = coupon, the bond
trades at par.
1200
1100
1000
800
0
0.01
0.02
0.03
0.04
0.05
0.06 0.07
6.375%
0.08
0.09
0.1
Discount Rate
When the YTM > coupon, the bond trades at a discount.
Interest Rate Risk
• Interest rate risk is the risk that arises for bond owners from
unexpected changes in interest rates.
➢ All other things being equal, the greater the time to maturity, the
greater the interest rate risk.
➢ All other things being equal, the lower the coupon rate, the
greater the interest rate risk.
Interest Rate Risk and Maturity
Consider a $1,000 face value bond with a 10% coupon rate
TIME TO MATURITY
Market Interest rate
5%
10%
15%
20%
1 year
30 years
$1,047.62
$1,000.00
$956.52
$916.67
$1,768.62
$1,000.00
$671.70
$502.11
Interest Rate Risk and Coupon
Consider two 30 year bonds (A and B), both have $1,000
face value. Bond A has coupon rate of 5% and Bond B has
a coupon rate of 10%.
COUPON RATE
Market Interest rate
5%
10%
15%
20%
Bond A
Bond B
$1,000.00
$528.65
$343.40
$253.16
$1,768.62
$1,000.00
$671.70
$502.11
Bond Concepts (summary)
1.
Bond prices and market interest rates move in opposite
directions.
2.
When coupon rate = YTM, price = par value.
When coupon rate > YTM, price > par value (premium bond)
When coupon rate < YTM, price < par value (discount bond)
3.
A bond with longer maturity has a higher relative (%) price
change than one with shorter maturity when the interest rate
(YTM) changes. All other features are identical.
4.
A lower coupon bond has a higher relative price change than a
higher coupon bond when the YTM changes. All other features
are identical.
Equity Valuation
What is a share?
Source
Share Valuation
• A share (or equity investment) is simply “part-ownership in a
company” and form parts of equity on the balance sheet.
A = L+ E
• In principle, shares can be valued in exactly the same way as
bonds by calculating the PV of all future CFs.
• However, share valuation is more difficult than bond
valuation for two reasons:
- Uncertainty of promised cash flows
- Shares have no maturity
Ordinary Share Valuation
• The value of a share is the present value of all
expected cash flows to be received from the share,
discounted at a rate of return (R) that reflects the
riskiness of those cash flows.
• The expected cash flows to be received from a share
are ALL future dividends.
• Dividend growth is an important aspect of share
valuation.
Ordinary Share Valuation
We will consider three cases:
(1) Zero Growth (or Preference share valuation)
(2) Constant Growth
(3) Variable Growth
(1) Zero Growth Valuation
• Shares have a constant dividend (D) into perpetuity, with no
growth in dividends. This implies that:
Div = Div0 = Div1 = Div2 = Div3 … …
• The value of a share (P0) is then the same as the value of a
perpetuity.
Div1
P0 =
rE
Where Div1 is dividend at t=0
rE is the cost of equity (the required rate of return)
Example 6: Preference Share Valuation
Assume Wave Industries is expected to pay a constant annual
dividend of $3 per share indefinitely. If the discount rate is 15
percent, what is the value of the share?
Div1 $3.00
P0 =
=
= $20.00
rE
0.15
(2) Constant Growth Valuation (DDM)
• Dividends grow at the same rate each time period, g,
forever.
Div t = Div 0  (1 + g)
n
• The value of a share (P0) is then the same as the value of a
growing perpetuity.
D t +1
D 0  (1 + g)
D1
Pt =
 P0 =
=
rE - g
rE − g
rE − g
• This is the Dividend Discount Model, also referred to as
Dividend Growth Model (an application of a growing
perpetuity)
Example 7: Constant Growth Valuation
Assume Alpha, Inc. has just paid an annual dividend of 15 cents per
share, which is expected to grow at 5% per annum forever. What
price should you pay for the share if the required rate of return on the
investment is 10%?
D0× (1 + g)
P0 =
rE − g
$0.15(1.05)
=
0.10 − 0.05
= $3.15
Important note: P0 values all future dividends from year 1 onwards
Components of Required Return
D1
P0 =
rE − g
D1
(rE − g) =
P0
D1
rE =
+g
P0
rE = dividend yield + capital gains yield
Components of Required Return
Revisit Example 7
What are the dividend yield and the capital gains yield in Example 7?
D1 = $0.15  (1 + 0.05) = $0.1575
D1
rE =
+g
P0
rE = 5% + 5% (thus, 5% Dividend Yield and 5% Capital Gains)
rE = 10%
(3) Variable Growth Valuation
• Allows for different growth rates.
• Dividends cannot grow at a rate above the required rate of return
indefinitely but can do so for a number of years.
• Dividends will grow at a constant rate at some time in the future.
P0 =
D1
(1 + rE )1
+
D2
(1 + rE ) 2
+ ... +
Dn
(1 + rE ) n
+
Pn
(1 + rE ) n
Example 8: Variable Growth Valuation
A company has just paid an annual dividend
of 15 cents per share and that dividend is expected
to grow at a rate of 20% per annum for the next 3
years and at a rate of 5% per annum forever after
that.
Assuming a required rate of return of 10%,
calculate the current market price of the share.
Example 8: continued
Variable Growth Valuation 4-step Approach
Initial growth period
0
1
D1
2
3
D2
D3
4
D4
t
…
Step 1:
Calculate the value of the dividends at the end of each year, Dn
(during the initial growth period) i.e. the first three years.
STEP ONE: calculations
YEAR
1
EXPECTED
DIVIDEND
D1 = $0.15(1.2)1 = $0.18
2
D2 = $0.15(1.2)2 = $0.216
3
D3 = $0.15(1.2)3 =$0.2592
Example 8: continued
Variable Growth Valuation 4-step Approach
0
1
D1
2
3
D2
D3
4
n
PV(D1)
PV(D2)
PV(D3)
Step 2: Find the PV of expected dividends during the initial
growth period
STEP TWO: calculations
Year
1
Expected
Dividend
$0.180
PV
PV0(D1) = $0.180 ÷ (1.10)1 = $0.164
2
$0.216
PV0(D2) = $0.216 ÷ (1.10)2 = $0.179
3
$0.259
PV0(D3) = $0.259 ÷ (1.10)3 = $0.195
Example 8: continued
Variable Growth Valuation 4-step Approach
Initial growth period
0
PV(D1)
1
2
3
D1
D2
D3
4
D4
PV(D2)
PV(D3)
Step 3: Find the value of the share at the
end of the initial growth period.
D4
P3 =
rE − g
n
…
STEP THREE: calculations
Note:
➢ The share reverts to its long run historical (constant) growth
rate in year 4.
➢ This means we can use the dividend growth model to calculate
the price at the end of the initial growth phase which occurs at the
end of time period 3.
D 3 (1 + g)
D4
P3 =
=
rE − g
rE − g
$0.259(1.05)
=
0.10 − 0.05
= $5.44
Example 8: continued
Variable Growth Valuation 4-step Approach
0
1
2
3
D1
D2
D3
4
D4
PV(D1)
PV(D2)
PV(D3)
PV(P3)
Step 4: Discounting & Summing Up
D4
PP33 =
rrE −−gg
n
…
STEP FOUR: Calculations
Determine the PV of the price found in step 3 and then
sum this to the PV of dividends found in step 2.
P3
$5.44
PV0 (P3 ) =
=
= $4.09
3
3
(1 + rE )
(1.10)
Variable Growth Valuation
Example 8 Solution continued
P0 =
D1
(1 + R)1
+
D2
(1 + R) 2
+
D3
(1 + R) 3
+
P3
(1 + R) 3
$0.180 $0.216
$0.259
$5.44
P0 =
+
+
+
(1.10)1 (1.10)2 (1.10)3 (1.10)3
P0 = $4.63
Variable Growth Valuation
Example 8 Solution continued
D1
D2
D3
P3
P0 =
+
+
+
(1 + R)1 (1 + R)2 (1 + R)3 (1 + R)3
$0.180 $0.216 $0.259
$5.44
P0 =
+
+
+
(1.10)1 (1.10)2 (1.10)3 (1.10)3
P0 = $4.63
NOTE: you could use the CFj function of the calculator to enter these dividends as
cash flows, recalling that CFj requires a value at time 0. Here, we have none, so we
will assign 0 as CF at time 0.
Input the following (remember to clear cash flow memory! How? Arrow up then C
then 0):
10 I/YR then 0 CFj then 0.18 CFj then 0.216 CFj then (0.259+5.44) CFj arrow down
PRC to get NPV of all dividends! Clear your memory again!
(note both 0.259 and 5.44 are at time 3 so add them as one single CF at time 3).
In the final exam, please first indicate workings using the equation though.
❑ The focus of Lecture 3 has been on valuation of debt securities
(bonds) and equity securities (shares)
❑ Next week, we turn our attention to capital budgeting and
familiarise ourselves with many of the evaluation techniques
often used in practice.
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