Topic 4: Valuation of Securities

Topic Overview
Lesson 4.1
4.1.1 Valuation of Financial Assets
4.1.2 Valuation of a Firm
4.1.3 Difference between debt & equity
4.1.4 Features of a Bond
4.1.5 Bond Yield
4.1.6 Short-term Debt Securities
4.1.7 Long-term Debt Securities
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 2
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4.1.1 Valuation
 The value of a financial asset is equal to the
present value of all future cash flows – this
principle will be used throughout this subject
and is the basis of financial decision-making
 Topic 2 provided the mathematical tools we
need to calculate the present value of a future
cash flow, or a series of future cash flows
 A critical element in these calculations is the
discount rate – a measure of the time value of
money – that we use to discount future cash flows
VALUATION OF FINANCIAL ASSETS
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 3
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4.1.1 Valuation
 Throughout this subject, different terms are
sometimes used to describe the discount rate
DISCOUNT RATES
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 4
Interest rate Yield Opportunity cost
Required rate of return Cost of capital
In theory these terms
describe different
things, because they
are arrived at in
different ways
In financial mathematical
terms, they are treated in the
same way – they are used to
discount future cash flows to
determine their value today
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4.1.1 Valuation
 If we are valuing a bank deposit, the
appropriate discount rate is the interest rate,
and this was the approach used in Topic 2
 For most investment opportunities, the discount
rate that best determines the value to us of a
future cash flow is the opportunity cost, which
is the rate of return we could get from the next
best alternative investment
 If an investment does not earn at least the
opportunity cost, it should not be undertaken
DISCOUNT RATES
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 5
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4.1.2 Valuation
 Consider
again the
balance
sheet from
Topic 1:
VALUATION OF A FIRM
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 6
 In theory we could determine the value of the
firm by calculating the total value of the
assets on the left-hand side OR the total
value of the liabilities and shareholders’
equity on the right-hand side
Assets Liabilities & Equity
Current assets Current liabilities
Long‐term assets Long‐term debt
Owners’ equity
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4.1.2 Valuation
 In practice, it is usually difficult to value the
assets of a firm, because there is usually not
a liquid market in which the value of the
assets can be determined
 It is generally easier to value the liabilities
and shareholders’ equity (i.e. the securities
issued by the firm, which represent claims on
the assets of the firm and the net income
generated by those assets) because they are
usually traded in liquid markets
VALUATION OF A FIRM
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4.1.3 Valuation
 There are fundamentally two different types
of securities issued by a firm – debt and
equity – and the differences between them
impact upon how we value them
DEBT v EQUITY
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 8
Debt
securities…
almost always have a limited life
usually pay a fixed rate of return
represent a contractual claim on the income and assets
of the firm which can be enforced in the courts
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4.1.3 Valuation
DEBT v EQUITY
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 For these reasons, debt is considered a less
risky form of investment than equity
Equity
securities…
normally have an unlimited life
pay a return which is often highly
variable (and may be negative if
there is a fall in share price)
represent a residual claim on the income and assets of
the firm – equity holders own whatever is left of the
income and assets after contractual claims are met
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Any instrument
issued by a firm or
government to borrow money
One year or less
to maturity
4.1.4 Debt securities
 The general term
for a debt security
is a bond
DEBT SECURITY TERMINOLOGY
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 10
Short‐term
More than one
year to maturity
Long‐term
Bond
A promise to repay interest
and/or the face value
according to specified terms
In practice, the
term “bond” is
used for longterm
securities
 We normally
classify debt
securities as:
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The amount that has
been borrowed (also
called the par value)
The length of time between
the valuation date (i.e.
today) and the maturity date
The present value of
all future cash flows
The date by which the
face value must be repaid
4.1.4 Debt securities
 There are key features of debt securities that
we need to understand in order to value them:
DEBT SECURITY TERMINOLOGY
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Face value Maturity date
Price
Term to maturity
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4.1.5 Debt securities
DEBT SECURITY TERMINOLOGY
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The yield… is also called the debt cost of capital
is the rate used to
discount the face
value to a present
value to get the
price of the security
is the realised
return if the
security is held
until maturity
represents the
opportunity
cost of capital
for the investor
Opportunity cost
This is related to the general level of
interest rates and the risk of the security
The return the investor would
receive from
an investment
of similar risk
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4.1.5 Debt securities
 The yield can change on a daily basis
(possibly because the risk of the security has
changed, but normally because of
fluctuations in interest rates)
 The price of the security can therefore
fluctuate over the life of the security
 If the investor sells the security prior to
maturity, the realised return will be a function
of the purchase price and selling price – we
will discuss realised return in detail in Topic 6
DEBT SECURITY TERMINOLOGY
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4.1.6 Debt securities
SHORT-TERM DEBT SECURITIES
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Short‐term
securities…
do not normally pay interest
only involve one cash flow (the
repayment of the face value)
are usually
referred to
as discount
securities
are sold at a price which is at a
discount to face value (to reflect
the time value of money)
are normally used by firms and governments to meet
day‐to‐day, short‐term and seasonal funding needs
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Essentially IOUs used by
firms to borrow money
4.1.6 Debt securities
 Examples of short-term debt securities:
SHORT-TERM DEBT SECURITIES
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities
Promissory notes
15
Also referred to as
commercial paper
Are unsecured; hence
only be used by firms
with a high debt rating
Similar to commercial paper,
but repayment of the debt is
guaranteed by a bank
Bank bills
Promissory notes issued by
the federal government for
short‐term funding (referred
to as Treasury bills in the US)
Treasury notes
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4.1.7 Debt securities
LONG-TERM DEBT SECURITIES
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Long‐term
securities…
are usually referred to using
the general term bonds
are sometimes
called notes
when they are
medium‐term
are sometimes
called debentures
if they are
unsecured
are normally issued by firms and
governments for long‐term funding
are
sometimes
issued in
perpetuity,
in which
case they
are called
consols
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Interest on the amount
that has been borrowed
4.1.7 Debt securities
 In most cases the issuer of a long-term bond
promises to make regular coupon payments
LONG-TERM DEBT SECURITIES
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities
Usually paid annually, semiannually
or quarterly Coupon rate
The annual rate of
interest (calculated as
a percentage of the
face value of the bond)
Note: Australian Treasury
bonds always make semiannual
coupon payments
Coupon payments
(or coupons)
17
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4.1.7 Debt securities
 The value of each coupon payment is given
by the following formula:
LONG-TERM DEBT SECURITIES
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For example, if a bond has a face value of
$1000 and a 7% coupon rate, the quarterly
coupon payments will be equal to 7% x
$1000 / 4 = $17.50
Coupon rate Face value
Number of coupon payments per year
CPN


(6.2)
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Topic Overview
Lesson 4.2
4.2.1 Price of Short-term Security
4.2.2 Yield of Short-term Security
4.2.3 Price of a Zero Coupon Bond
4.2.4 Yield of a Zero Coupon Bond
4.2.5 Price of a Coupon Bond
4.2.6 Discount and Premium Bonds
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4.2.1 Valuation of short-term securities
 As the only future cash flow is the payment
of the face value, the price of a short-term
security is the face
value discounted
over the remaining
life of the security:
PRICE OF A SHORT-TERM SECURITY
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where P = the price of the security
FV = the face value of the security
y = the annual yield on the security
d = the number of days to maturity
1
365
P FV
y d

    
 
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4.2.2 Valuation of short-term securities
 Using the formula on the previous slide gives
us the price if we know the yield, but in
practice it is often the price that is observable
and we can use the price to determine the
yield
 The previous
formula can
be rearranged
to solve for
the yield:
YIELD OF A SHORT-TERM SECURITY
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 21
y FV 1 365
P d
       
  
y = the current yield
FV = face value, P = price
d = days to maturity
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4.2.3 Valuation of bonds
 The simplest type of bond is one that does
not make coupon payments
 In other words, the coupon rate is zero, so
this is referred to as a zero-coupon bond
 Because of the time value of money, the price
of a zero-coupon bond (prior to maturity) will
be heavily discounted, so this is also referred
to as a discount security or a pure
discount bond
PRICE OF A ZERO-COUPON BOND
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4.2.3 Valuation of bonds
 As the only future cash flow is the payment
of the face value, the formula for the price of
a zero-coupon bond is
essentially Formula 3.2
(the present value of a
single cash flow):
PRICE OF A ZERO-COUPON BOND
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where P = the price of the bond
FV = the face value of the bond
y = the yield on the bond
n = the number of periods
1 n
P FV
y


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4.2.4 Valuation of bonds
 Again, if we know the yield we can determine
the price, but in practice the price that is
observable and we can use that to determine
the yield
 We can rearrange
the previous formula:
YIELD OF A ZERO-COUPON BOND
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1 n
P FV
YTM


to solve for the YTM
 This is called the yield to maturity, because
it will be realised if the bond is bought at the
current price and held to maturity)
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4.2.4 Valuation of bonds
 The formula for the yield to maturity of a
zero-coupon bond is:
YIELD OF A ZERO-COUPON BOND
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where YTM = the yield to maturity
FV = the face value of the bond
P = the price of the bond
n = the number of periods
1/
1
n YTM FV
P
     
 
(6.2)
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4.2.5 Valuation of bonds
 The cash flows from a coupon bond (a bond
with a non-zero coupon rate) consist of the
periodic coupon payments and the payment
of the face value on maturity
 As with all securities, the price of the bond is
the present value of all future cash flows
 If the coupon rate is fixed (which we will
assume to be the case in this subject) the
coupon payments remain constant and
hence constitute an annuity
PRICE OF A COUPON BOND
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 26
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4.2.5 Valuation of bonds
 The formula for the price of a coupon bond is
a combination of the formulas for the present
value of an annuity and a single cash flow:
PRICE OF A COUPON BOND
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where CPN = the coupon payment amount
P = bond price, y = the yield per period
n = number of periods, FV = face value
   
1 1 1
1 1 n n
P CPN FV
y y y
 
     
     
(6.3)
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4.2.6 Valuation of bonds
 An examination of Formula 6.3 shows that
there is an inverse relationship between the
price of a bond and its yield
 This is not only true mathematically, but
makes intuitive sense – the higher the yield,
or required rate of return (i.e. the higher the
rate at which you discount future cash flows),
the lower the present value of the future
cash flows and hence the lower the price of
the bond, and vice versa
DISCOUNT AND PREMIUM BONDS
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4.2.6 Valuation of bonds
 If we calculate the price of a bond for various
yields, we can see a relationship between the
yield, coupon rate, price and face value
 If the yield equals the coupon rate, interest is
paid at the required rate of return, and the
bond trades at a price equal to its face value
 If the yield is less than the coupon rate (i.e.
interest payments are higher than the
required rate of return), this makes the bond
more attractive and price exceeds face value
DISCOUNT AND PREMIUM BONDS
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4.2.6 Valuation of bonds
 If the yield exceeds the the coupon rate (i.e.
interest payments are less than the required
rate of return), this makes the bond less
attractive and price is less than face value
 We can summarise the relationship between
the characteristics of a bond as follows:
DISCOUNT AND PREMIUM BONDS
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If… then… and the bond is …
Yield < Coupon Rate Price > Face Value at a Premium
Yield > Coupon Rate Price < Face Value at a Discount
Yield = Coupon Rate Price = Face Value at Par
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Topic Overview
Lesson 4.3
4.3.1 Ordinary Shares
4.3.2 Preference Shares
4.3.3 Hybrid Securities
4.3.4 Dividend Yield and Capital
Gain Yield
4.3.5 General Dividend Discount
Model
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4.3.1 Equity securities
 Unlike a debt-holder (who has lent money to
the firm), a shareholder has invested in, and
is a part-owner of, the firm
 The shareholder’s proportional level of
ownership is based on the number of shares
held
 There are two main types of shares:
TYPES OF SHARES
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Ordinary shares & Preference shares
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4.3.1 Equity securities
ORDINARY SHARES
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Ordinary shares… are the most commonly traded
securities in Australia
have the right to dividends
have the right to
attend the AGM, elect
the Board of Directors
and vote on major
proposals (such as
takeovers and mergers)
Ordinary
shareholders…
have the right to receive their
share of the net assets of the
firm if it is wound up
This is a residual claim – shareholders
own what is left after
meeting all contractual claims
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If dividends cannot be
paid in a given year, they
carry forward and must
be paid in future years
before ordinary
dividends can be paid
4.3.2 Equity securities
PREFERENCE SHARES
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Preference shares…
give shareholders preference
over ordinary shareholders,
in the payment of dividends
and the distribution of the
firm’s assets if it is wound up
are usually cumulative
Ordinary shareholders can’t
receive a dividend unless
preference shareholders
have received their dividends
normally pay a fixed dividend
Cumulative shares
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4.3.3 Equity securities
 Convertible bonds and preference shares are
examples of hybrid securities – securities
that have features of both equity and debt
HYBRID SECURITIES
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 35
Convertible
bonds
Are like equity,
in that…
While ordinary
bond prices
respond to interest rates, convertible
bond prices are normally linked to the
value of the underlying shares
Are technically
debt…
They rank ahead of all shareholders in the payment of interest
and distribution of the net assets of the company.
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4.3.3 Equity securities
 Convertible bonds and preference shares are
examples of hybrid securities – securities
that have features of both equity and debt
HYBRID SECURITIES
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 36
Preference
shares…
Are like debt,
in that…
They have
priority over
ordinary
Are technically shareholders
equity…
They normally
pay a fixed
amount
They rank behind
debt holders
Unlike debt holders they cannot force the
firm into bankruptcy if they aren’t paid
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4.3.4 Valuation of shares
 As noted previously, the value of a security is
the present value of all future cash flows
 In the case of debt securities, those cash
flows are discounted at the yield or debt cost
of capital, which reflects the general level of
interest rates and the risk of the bond
 For an equity security, the discount rate is the
equity cost of capital (rE) – also called the
required rate of return – the return available
from other investments of similar risk
EQUITY COST OF CAPITAL
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 37
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4.3.4 Valuation of shares
 In the case of a share, the future cash flows
consist of:
 Dividends received while you hold the share
 The price received when you sell the share
 For a one-year investor, the price of a share
is given by:
ONE-YEAR INVESTOR
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where
Div1 = Year 1 dividend
P1 = Selling price at the end of year 1
1 1
0 1 E
P Div P
r



(7.1)
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4.3.4 Valuation of shares
 We can rearrange Formula 7.1 as follows:
DIVIDEND YIELD AND CAPITAL GAIN
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 This enables us to break down the total
return (rE) into two components – the
dividend yield
Dividend
yield
Capital
gain yield
and the capital gain yield
1 1 1 1 0
0 0 0
1 E
r Div P Div P P
P P P
 
    (7.2)
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4.3.5 Valuation of shares
 If the one-year investor
sells to another one-year
investor, the value of the
share to that investor is:
MULTI-YEAR INVESTOR
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 40
2 2
1 1 E
P Div P
r



 Substituting this into Formula 7.1 we get:
 
1 1 1 2 2
0 1 1 1 1 2 E E E E
P Div P Div Div P
r r r r

   
   
(7.3)
 This is identical to the value of the share
from the point of view of a two-year investor
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4.3.5 Valuation of shares
 We can see that the price at which any
investor can expect to sell a share will be the
value placed on it by the purchaser
 The value to the purchaser will be the present
value of the future cash flows expected to be
received by that investor, which consist of:
 Dividends received while that investor holds
the share
 The price received when that investor sells the
share
MULTI-YEAR INVESTOR
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 41
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4.3.5 Valuation of shares
 We can continue this process for any number
of years by replacing the final share price with
the value that the next purchaser will be
willing to pay
 This concept is embodied in the general
dividend-discount model:
GENERAL DIVIDEND-DISCOUNT MODEL
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 42
   
1 2
0 2 …
1 1 1
n n
n
E E E
P Div Div Div P
r r r

   
  
(7.4)
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4.3.5 Valuation of shares
 Assuming the share exists forever, the value
of a share at any time is equal to the present
value of an infinite stream of future dividends
GENERAL DIVIDEND-DISCOUNT MODEL
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 43
  0
1 1
t
t
t E
P Div
r



   This can also be
expressed as:
   
1 2 3
0 2 3 …
1 E 1 E 1 E
P Div Div Div
r r r
   
  
(7.5)
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Topic Overview
Lesson 4.4
4.4.1 Valuation of Preference Shares
Valuation of Ordinary Shares
4.4.2 – Scenario 1
4.4.3 – Scenario 2
4.4.4 – Scenario 2 (Worked example)
4.4.5 – Scenario 3
4.4.6 – Scenario 3 (Worked example)
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4.4.1 Valuation of shares
 Formula 7.5 embodies the concept of the
dividend-discount model, but it is clearly not
practical to separately calculate the present
value of an infinite stream of dividends
 In order to implement the model, we need to
make some assumptions about the future
pattern of dividends
 Typical assumptions are that dividends will:
 Remain constant
 Grow at a constant rate
CONSTANT DIVIDENDS
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 45
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4.4.1 Valuation of shares
 If dividends are expected to remain constant,
the infinite steam of dividends constitutes a
perpetuity, and we can use the formula for
the present value of a perpetuity:
CONSTANT DIVIDENDS
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 46
where P0 = the price of the share
Div = the constant dividend
rE = the equity cost of capital
0
E
P Div
r

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4.4.2 Valuation of shares
 If dividends are expected to grow at a
constant rate, we can use the formula for the
present value of a growing perpetuity:
CONSTANTLY GROWING DIVIDENDS
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where P0 = the price of the share
Div1 = the next dividend
rE = the equity cost of capital
g = the constant growth rate
1
0
E
P Div
r g

 (7.6)
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4.4.3 Valuation of shares
 Sometimes we can make the assumption of
constantly growing dividends some time in
the future, but in the meantime dividends are
expected to follow some other pattern (e.g.
grow at a different rate)
 The only way to solve this is to determine the
interim dividends, separately calculate the
present value of each interim dividend (using
Formula 3.2) and then value the constantly
growing dividends that will occur in the future
VARIABLE GROWTH RATES
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4.4.3 Valuation of shares
 A generic formula for this calculation is:
VARIABLE GROWTH RATES
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 49
 Note that:
 n is the number of dividends before dividends
begin to grow at constant growth rate g
 the n+1th dividend is the first dividend of the
growing perpetuity
   
1
0
1
1
1 1
n
t n
t n
t E E E
P Div Div
r r g r


  
   
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4.4.3 Valuation of shares
VARIABLE GROWTH RATES
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   
1
0
1
1
1 1
n
t n
t n
t E E E
P Div Div
r r g r


  
   
 The first part of this formula gives the sum of
the present values of the first n dividends
 The second part gives the value of all
dividends from n+1 onwards, as at period n
 This value therefore must be discounted at rE
for n periods to find its value at time 0
La Trobe Business School
Lecture Example 4.1
What is the price of a $5000 zero-coupon bond
with 4 years to maturity and a yield of 6% p.a.?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 51
Appendix
2ND FV
4 N
6 I/Y
5 0
CPT PV
0 0 FV
This first step clears all TVM values from previous entries.
For the remaining examples shown in this appendix, it is
assumed that you have (a) cleared the TVM values as
shown here, (b) explicitly entered 0 for variables not used,
or (c) intentionally left some TVM values unchanged
because you are carrying out a
series of similar calculations (e.g.
see Lecture Example 4.6).
La Trobe Business School
Lecture Example 4.2
What is the price of a 4-year $5000 bond with
9% semi-annual coupons and a yield of 7% p.a.?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 52
Appendix
8 N
3 .
2 2
5 0
CPT PV
0 0 FV
5 I/Y
5 PMT
This is the first Lecture
Example we have done
that uses all 5 TVM
variables, and therefore
there is no need to clear
the TVM variables before
performing this
calculation.
La Trobe Business School
Lecture Example 4.3
What is the yield to maturity of a $2000 bond
with 7 years to maturity and 6% annual coupons,
and which is trading at $2168?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 53
Appendix
7 N
2 1
1 2
2 0
CPT I/Y
0 0 FV
6 8
0 PMT
+/– PV
Either the present value, or the
payments and the future value,
need to be negative –
otherwise you will get an error.
(The answer is 4.57%.)
La Trobe Business School
Lecture Example 4.4
What is the value of a 4-year $5000 bond with 9%
semi-annual coupons, if the yield is 9% and 11%.
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 4 – Valuation of Securities 54
Appendix
8 N
4 .
2 2
5 0
CPT PV
0 0 FV
5 I/Y
5 PMT
5 . 5 I/Y CPT PV
Tip – If you need to carry out a series
of similar calculations, and you know
what has previously been stored in
each TVM variable, you need only
enter the altered variables to do the
next calculation, as shown here.